Math 9 Chapter 5 Polynomials PDF

Section 5.1 - Defining Polynomials ♦ 157 West Point Grey Academy 5.1 Variable A variable is a letter whose value is an unknown real number. Examples: a, b...

Section 5.1 - Defining Polynomials ♦ 157 West Point Grey Academy 5.1 Variable A variable is a letter whose value is an unknown real number. Examples: a, b, c, f , x, y, z Term A term is a number, or product of a number and variable(s) raised to a power. Examples: 5 , 2x , - 3x2 , - 2xy Coefficient The coefficient of a term is the numerical factor. Examples: Term Coefficient 5x 5 x2 1 3 3 - 1.3xy - 1.3 -y -1 -3 -3 Note: If a term is a number only, it is called a constant term or simply a constant. A letter that can be any one of various numbers is called a variable. For example in the term 2x3 , 2 is the coefficient, x is the variable, and 3 is the exponent. Monomial A monomial is an expression of the type ax n , where a is a real number constant, and n is a non-negative integer. Examples of monomials: 3 y2 , - 4 , 2x 4 Examples of non-monomials: 1 , x , x 3 2 x Polynomial A polynomial is a monomial, or a combination of sums and/or differences of monomials. Examples: 3x , 2x - 1, 4x2 + y + 2 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 158 ♦ Chapter 5 - Polynomials West Point Grey Academy Degree and Leading Term of a Polynomial The degree of a term is the sum of the exponents of its variables. The leading term of a polynomial is the term of highest degree. The degree of a polynomial is the same as the degree of its leading term. Consider the following polynomials: 5x2 - 3x + 4 Term 5x2 - 3x 4 Degree 2 1 0 Leading Term 5x 2 Degree of Polynomial 2 - 5 + xy Term −5 xy Degree 0 2 Leading Term xy Degree of Polynomial 2 The following are names for certain kinds of polynomials Type Definition Examples Monomial A polynomial of one term 3 - 2x 5x2 - 4xy 0 22 x2 Binomial A polynomial of two terms 2x - y 3x - 1 2xy - x2 Trinomial A polynomial of three terms x 2 - 3x + 4 5y2 - 2ay - a2 Note: A polynomial is usually written in descending order of powers. For example, the polynomial 2x + 4 - 3x2 is written as - 3x2 + 2x + 4. A polynomial with more than one variable is written in alphabetic order. For example, the polynomial xy + y 2 - x 2 is written as - x 2 + xy + y 2 , since all terms are of degree two, and x is before y in alphabetic order. Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.1 - Defining Polynomials ♦ 159 West Point Grey Academy Algebra Tiles positive x2 negative x2 positive xy negative xy positive x negative x positive y negative y postive 1 negative 1 Example 1 Express the polynomial 2x2 - 3x + 2 using algebra tiles. ►Solution: Example 2 Express the polynomial - x2 + 2x - 3 using algebra tiles. ►Solution: Example 3 Express the polynomial - 2xy + 3x - 2y + 1 using algebra tiles. ►Solution: Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 160 ♦ Chapter 5 - Polynomials West Point Grey Academy Like Terms Terms with the same variable raised to exactly the same powers are called like terms. Terms that are not like terms are called unlike terms. Like Terms Unlike Terms 5x, 3x 5x 2, 3x 2x 2, - x 2 3x 2, 3y 2 2xy, - 4yx 2xy, - 3yz xy 2, y 2 x xy 2, x 2 y Like terms can be added or subtracted, but unlike terms cannot be added or subtracted. Note: xy is the same as yx Example 4 Simplify 4x2 + 3x2 - 2x2 ►Solution: 4x2 + 3x2 - 2x2 = 5x2 Example 5 Simplify 4x + 3x2 - 2x2 ►Solution: 4x + ^3x2 - 2x2h = x2 + 4x ( 4x is a unlike term to 3x2 and 2x2 ) Example 6 Simplify 2xy 2 + 3x 2 y - 4x 2 y + xy 2 ►Solution: ^3x 2 y - 4x 2 y h + ^2xy 2 + xy 2 h =- x 2 y + 3xy 2 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.1 - Defining Polynomials ♦ 161 West Point Grey Academy 5.1 Exercise Set 1. Match the description on the right with the expression on the left. (Some may have two descriptions) a) 2 i) monomial x b) 2 - x ii) binomial c) xy2 iii) trinomial d) - 5xy iv) polynomial with leading coefficient of − 4 e) 6y - 4x2 v) non-polynomial f) x 2 - 3x + 5 vi) polynomial of degree higher than 2 g) x2 - xy + y + 3 vii) polynomial written in descending power of x h) xy + y - x 2. Determine if the expression is a polynomial. a) x0 y/n b) 1 y/n x c) 2x2 + 3 y/n d) 2xy + y - 3 y/n e) x + 2x y/n f) 1 y/n 3 g) 2 x 2 + 3x - 1 y/n h) x2 + x + x0 y/n i) x- 2 + x- 1 + x y/n j) 2- 3 x2 y/n 3. Determine the degree of each polynomial. a) 3x - x2 + 2 b) 23 + x2 c) - 5 d) 32 e) 23 x - xy f) - y + 3x2 g) 7x2 + 25 h) 2 - x + 5x2 i) x0 j) 2 x 2 + 3x Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 162 ♦ Chapter 5 - Polynomials West Point Grey Academy 4. Determine the coefficient of each monomial. a) 2x b) - 8x3 c) 1.9y5 d) 2 x11 3 e) - 7 x6 f) 3.7x0 2 g) - 8 h) 0 5. Arrange in descending order of powers. a) xy - x2 + y2 b) 2x - xy + x2 c) 22 + x + x2 d) 50 - x2 + x e) y2 - 2xy + x2 f) xy - 8x2 - y 6. Write a polynomial expression for the algebraic tile models. a) b) c) d) Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.1 - Defining Polynomials ♦ 163 West Point Grey Academy 7. Draw algebra tiles to model the polynomials. a) - 2x2 + 2 b) 2x2 - 3x c) - 2x2 - x + 3 d) - 4x + 1 e) x2 + 4x - 2 f) 3x 2 - x + 2 g) - 2x 2 + 2x - 1 h) xy + x - y i) 2x 2 - xy + 3x - 2y j) - x 2 + xy + 2 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 164 ♦ Chapter 5 - Polynomials West Point Grey Academy 8. Simplify. a) 2x - 3 - x + 2 b) - 3 + 3x - 2 - x c) 2x2 + x - x2 - x d) 3x2 - y2 + 3y2 - x2 e) - 4x2 - 3x + 4 + 2x f) - 5x + 2y + 4y - x g) 4x - 3y + 6x - 5x h) - 2x2 - 5 + 2x + x2 - 3 - x i) 3xy - 4yx + 2xy - yx j) - 5xy - 2yx - y + 2x - 6yx k) 13x - 7 - 4x + 2 l) 22 + 5x + 7y - x + y - 28 m) 2 x + 1 x n) 2y + 3 y + y 3 3 4 o) 1 x + x - 5x p) - 8 + 11x - 5y + 7x - 7y + 6 2 q) 8x - 5x + 4 + 3y - 2x - 7 r) - 3x 2 - 4x + x 2 - 2x + 4 s) - 2xy + 3x + 5xy - 2y - x - y t) 4xy - 2x - 5xy + 4x - 7 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.1 - Defining Polynomials ♦ 165 West Point Grey Academy 9. Write a polynomial expression in simplified form for the algebra tiles. a) b) c) d) e) f) Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 166 ♦ Chapter 5 - Polynomials West Point Grey Academy 10. Simplify each polynomial by collecting like terms. a) - a + 3 + 15a2 - a - 1 - 3a2 b) 3 + 2b2 - 5b - 2b + 4b2 4 2 c) - 6c2 + c - 5c + 7c2 + 1 d) - 2d - 2d - 2d + d2 - 5d2 e) 2e - 5 + 4e2 + e + 1 - 2e f) - 3fg + 4gf - 4f2 + g2 6 3 g) 1 h2 5 1 h2 2h 9 h) 1 i2 + 2i - 1 i2 + 4 - 9 4 - +2 - - 3 6 i) 1 j2 j 1 j 3 j2 5 j2 j) 1 k2 1 2k 1 3 k 2 2k 3 4 - - 6 + 8 + 16 5 + 5 - + 10 - 15 + - 10 k) 1 x2 + 3 x + 1 x2 - 5 x l) 2 x2 - 1 x + 4 x2 + 1 x 4 4 6 6 5 3 3 5 m) 2.8x 2 - x + 1.4x 2 - 3.2x + 1 n) - 4.9x - 3.2y - 1.3x + 4.2y + 1 o) - 3.2x + 1.7y - 1.2x - 2.8y - 5 p) 2.9x + 2.3y - 1.9x - 1.8y q) 11 x + 2 y - 3 x - 1 y + 10 r) 11 x + 9 y - 2 x - 3 y - 37 4 3 5 3 2 5 3 10 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.2 - Adding and Subtracting Polynomials ♦ 167 West Point Grey Academy 5.2 Adding and Subtracting Polynomials Adding Polynomials Horizontally To add polynomials, we use a plus sign and add like terms. Arranging like terms together may save some computational errors. Example 1 Add ^- 2x2 - 4h + ^3x2 - 2x + 2h ►Solution: ^- 2x2 - 4h + ^3x2 - 2x + 2h - 2x2 - 4 + 3x2 - 2x + 2 - 2x2 + 3x2 - 2x - 4 + 2 x2 - 2x - 2 Adding Polynomials Vertically To add polynomials vertically, place the polynomials with like terms in the same columns. Leave a blank space if a term in one of the polynomials does not have a match in the other. Example 2 Add ^- 2x2 - 4h + ^3x2 - 2x + 2h ►Solution: - 2x2 -4 2 3 x - 2x + 2 x 2 - 2x - 2 Adding Polynomials with Algebra Tiles Adding polynomials with algebra tiles is similar to adding polynomials horizontally. Example 3 Add ^- 2x2 - 4h + ^3x2 - 2x + 2h ►Solution: = x 2 - 2x - 2 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 168 ♦ Chapter 5 - Polynomials West Point Grey Academy Subtracting Polynomials Horizontally To subtract a polynomial, we change the sign of every term. This is the same as multiplying by − 1. For example -^- 2x2 + 4x + 1h is written as 2x2 - 4x - 1 without brackets. Example 4 Subtract ^2x2 - 4x - 3h - ^- x2 + 2x - 1h ►Solution: ^2x2 - 4x - 3h - ^- x2 + 2x - 1h 2x2 - 4x - 3 + x2 - 2x + 1 2x2 + x2 - 4x - 2x - 3 + 1 3x2 - 6x - 2 Subtracting Polynomials Vertically To subtract a polynomial vertically, place the polynomials with like terms in the same columns in descending order of powers. Multiply the negative sign outside of the parenthesis into the polynomial, and add the polynomials. Leave a blank space if a term in one of the polynomials does not have a match in the other. Example 5 Subtract ^2x2 - 4x - 3h - ^- x2 + 2x - 1h ►Solution: 2x2 - 4x - 3 2x2 - 4x - 3 " -^- x2 + 2x - 1h + x2 - 2 x + 1 2x2 - 4x - 3 + x 2 - 2x + 1 3x2 - 6x - 2 Subtracting Polynomials with Algebra Tiles To subtract a polynomial with algebra tiles, the signs of the polynomial tiles with a negative in front of the parentheses need to be changed before combining the tiles. Example 6 Subtract ^2x2 - 4x - 3h - ^- x2 + 2x - 1h ►Solution: = 3x2 - 6x - 2 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.2 - Adding and Subtracting Polynomials ♦ 169 West Point Grey Academy 5.2 Exercise Set 1. Add. a) ^2x - 3h + ^- 4x + 1h b) ^- 3x2 + 2xh + ^ x2 - 3x - 2h c) ^ x2 - 2h + ^ x2 + 2h d) ^2x - 3h + ^- x2 - 3x + 1h e) ^5 - x2 + 2xh + ^- 3x - 2x2 + 1h f) ^- 3x + 2 - 4x2h + ^- 4 + 2x2h g) ^- x 2 + 2 - 3x h + ^- 4x 2 + x - 5 h h) ^- 3 + 4x 2 - 2x h + ^5x - 2x 2 + 4 h i) ^4x - 2x 2 h + ^- 5 + x 2 h j) ^ 2 - 3x 2 + 4x h + ^ - x 2 - 2x h k) ^- 3xy - x 2 - 2y 2 h + ^- 2xy + x 2 - y 2 h l) ^ x 2 - y 2 - 4xy h + ^3xy - 3x 2 + 2y 2 h m) ^- 2xy + x2 - 3y2h + ^- y2 - xy + 2x2h n) ^3xy - x + yh + ^- 3y + 2x - xyh o) ^3x - 2xy + 2yh + ^ xy - 3yh + ^- 3y - xh p) ^- 2y + 3x + xyh + ^2xy - x - yh + ^- x - 4xyh Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 170 ♦ Chapter 5 - Polynomials West Point Grey Academy 2. Add. a) - 3x + 1 b) - 3x 2 + 4x 2x - 3 x 2 - 3x - 2 c) x 2 - 3 d) - x 2 - 2x + 3 x2 + 3 4x - 1 e) x 2 + 4x - 3 f) - 2x 2 -3 - 2x 2 - 2x - 1 4x 2 - 3x + 1 g) - x 2 - 4x + 2 h) 3x 2 - 2x - 5 - 4x 2 + x + 5 - 2x 2 + 5x - 3 i) 3 - 2x 2 + 5x j) - 2xy - 3x 2 + 4y 2 - x 2 + 2x 4xy + x 2 - y 2 k) 2x 2 + y 2 - 4xy l) 4xy + x - y - 3xy - 3x 2 + 2y 2 2x - xy + y m) 2x 2 - 3x + 2 n) xy - x + y - x + 4 - 2x 2 - 3x + y + 2xy 3 - x2 + x - y - 5xy - 2x Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.2 - Adding and Subtracting Polynomials ♦ 171 West Point Grey Academy 3. Add by algebra tiles. a) ^- 2x2 + x - 3h + ^ x2 - 2x - 1h b) ^- 3x - x2 + 4h + ^- 2 + 2x2 - 4xh c) ^- 4x + 2h + ^- x2 + x - 3h d) ^ x 2 - 3x h + ^2x 2 + x - 2 h e) ^- x 2 + 4x + 3 h + ^- 2x 2 - 2x - 1 h f) ^- x 2 - x h + ^2x 2 + 3 h + ^- 2x + 2 h g) ^ x 2 - 2 h + ^2x + 3 h + ^- 2x 2 - x h h) ^ xy - y + 2xh + ^- 2xy + 2y - xh Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 172 ♦ Chapter 5 - Polynomials West Point Grey Academy 4. Subtract. a) ^4x + 2h - ^- 2x + 3h b) ^- 5x + 2h - ^ x2 + x - 3h c) ^- x2 + 5 - 4xh - ^6 - 2xh d) ^- 2x2 + 3x - 1h - ^- x2 + x + 3h e) ^- 2x + x2 - 4h - ^3 - 2x2 - 4xh f) ^- 4x2 + 2xh - ^- 2x - 3x2 + 2h g) ^- 3x + 2x2h - ^- 3x + 2x2 - 1h h) ^5x + 4y - 9z h - ^- 11x + 9y - 5z h i) ^2x 2 + 34y h - ^7x 2 + 12y + 15z h j) ^- 5x - 2y + 3z h - ^- 2x + 9y h k) ^- 3x - 6y h - ^9x - 4y - 25z h l) ^- 2x - 3y h - ^4x + 2y h - ^ x - 3y h m) ^4x - 2y h - ^2x + 3yh - ^- 2x + y h n) ^- 2xy + y - 3xh - ^- y - 2x + xyh o) ^3xy + x - 2yh - ^ y - x - xyh p) ^ x2 - 3xy + y2h - ^- 2xy + x2 - y2h Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.2 - Adding and Subtracting Polynomials ♦ 173 West Point Grey Academy 5. Subtract. a) 5x + 2 b) - x 2 + 2x - 4 - 3x + 1 - 5x + 1 c) - 3x + 4 d) - 3x 2 +2 2 2 -x + x - 2 x - 3x e) - 2x 2 - x + 1 f) - 3x 2 - 2x - 4x 2 + 3x - 2 4x - 3 g) 3x 2 - 4x + 5 h) - x 2 -4 - x 2 - 3x + 3 - 3x + 2 i) x 2 - 3xy - y 2 j) 2x 2 - 5 + 3x x 2 + 3xy - y 2 - 2x + x 2 - 3 k) - x 2 - xy + y 2 l) - 3xy + x - 4y - 2xy + 3y 2 - 4x 2 - y - xy + 2x m) 4x - 3xy + 2y n) - 4y + 2xy - 2y + xy - 4x 2x - 3y - 4xy Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 174 ♦ Chapter 5 - Polynomials West Point Grey Academy 6. Subtract by algebra tiles. a) ^- 2x + 3h - ^- 4x - 1h b) ^- x 2 + 2 h - ^ x - 3 h c) ^ x2 - 4 - 3xh - ^2x - 1 + 2x2h d) ^- 3x2 - 2 + xh - ^- x2 + 2x - 1h e) ^3x 2 - x + 1 h - ^2x - x 2 h f) ^- 2x + x 2 - 1 h - ^2x 2 - 3 h g) ^- 2x 2 + 3x h - ^ x 2 - x + 1h h) ^2xy - y + 2x h - ^ xy + x - y h Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.2 - Adding and Subtracting Polynomials ♦ 175 West Point Grey Academy 7. Simplify the combined operations. a) ^2x - 3h - ^4x + 2h + ^3x - 4h b) ^ x2 - 1h + ^2x2 + 3h + ^3x2 - 4xh c) -^2 - x2 + xh + ^ x - 3x2 + 4h - ^2x2 - 1h d) ^- 2x - 1h - ^ x2 - 3x + 2h - ^- 3 - x2h e) ^- 3x2 + 4x - 6h - ^- 2x2 - 4h + ^5x - 3h f) ^- 4 + x2 + xh - ^- 6 - x + 3x2h - ^- x2 + xh g) ^- y2 - 7y + yh + ^- 2y2 + 5y - 2h - ^- 6y2 + yh h) ^2xy - x - yh - ^- 3xy + yh + ^- x + 2yh i) ^- x - xy - yh - ^2xy - yh - ^ x - yh j) ^- 3x + 2y - xyh + ^- 2xy - yh - ^ x - yh k) ^- 3x + 4y h + ^6x - 5yh - ^2x + 11y - 5z h l) ^- 2x 2 + 3xy - 1 h - ^- 2xy + x 2 - 1 h + ^ x 2 - 5xy h m) ^- 2xy + 9z h + ^4x 2 - 11z h - ^6x 2 + 8xy - 11z h n) ^3x 2 + 5y - 6z h - ^- 4x 2 - 7y + 2z h - ^ x 2 - 3y - 4z h Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 176 ♦ Chapter 5 - Polynomials West Point Grey Academy 8. Determine the perimeter of each figure. Simplify. (All lengths are in cm) a) b) 2x + 1 2+ 3x c) d) x 2x + 4 4x + 3 e) f) x+2 2x + 3 g) h) 2x r+1 x Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.2 - Adding and Subtracting Polynomials ♦ 177 West Point Grey Academy 9. Determine the perimeter. Simplify. (All lengths are in cm) 6 5a a) b) x a 3 3a 4 x 3 a 3x 4 2a 2 c) r 1 d) 2 3 + + x 12 x e) f) r r 4 x 6 6 x 12 g) h) x x 3 13 4 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 178 ♦ Chapter 5 - Polynomials West Point Grey Academy 10. Find the surface area of each figure. (All lengths are in cm) a) b) x 5 x 3 2 x c) d) 3 4 3x 2 3 x e) r f) x x 4 4 g) h) x 1.5 x 6 8 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.3 - Multiplying Polynomials ♦ 179 West Point Grey Academy 5.3 Multiplying Polynomials Multiplying Monomials To multiply two monomials, multiply the numerical factors, then multiply the variable factors. Example 1 Multiply ^2x2h^- 3xh ►Solution: ^2x2h^- 3xh = ^2h^- 3h^ x2h^ xh =- 6x2 + 1 =- 6x3 To find the product of a monomial and a polynomial with more than one term, we use the distributive property. Recall 2^3 + 4h = 2 # 3 + 2 # 4 = 6 + 8 = 14. The general form of the distributive property is given by a^ b + ch = ab + ac. This suggests the following rule: Multiplying Polynomials by Monomials To multiply a polynomial with more than one term by a monomial, use the distributive property to remove parentheses, then simplify. Example 2 Multiply - 4^3 - xh ►Solution: - 4^3 - xh =- 4 # 3 + ^- 4h # ^- xh =- 12 + 4x Example 3 Multiply - 4x ^ x 2 - 3x + 1 h ►Solution: - 4x ^ x 2 - 3x + 1 h = ^- 4x h^ x 2 h + ^- 4x h^- 3x h + ^- 4x h^1 h =- 4x 3 + 12x 2 - 4x Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 180 ♦ Chapter 5 - Polynomials West Point Grey Academy Multiplying Polynomials Using Algebra Tiles Example 4 Multiply - 3x^2x - 1h using algebra tiles. ►Solution: Therefore - 3x^2x - 1h =- 6x2 + 3x. Example 5 Multiply ^- 3x + 2h^2x - 1h using algebra tiles. ►Solution: Therefore ^- 3x + 2h^2x - 1h =- 6x2 + 7x - 2. Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.3 - Multiplying Polynomials ♦ 181 West Point Grey Academy 5.3 Exercise Set 1. Find each product. a) - 5 # 3 b) - 5 # 3x c) - 5x # 3x d) - 5x # 3x2 e) - 5x2 # 3x2 f) - 4 # ^- 7h g) - 4 # ^- 7xh h) - 4x # ^- 7xh i) - 4x # ^- 7x2h j) - 4x2 # ^- 7x2h 2. Find each product. a) 2^ x + 3h b) 2^ x - 3h c) - 2^ x + 3h d) - 2^- x - 3h e) 2x^ x + 3h f) 2x^ x - 3h g) - 2x^ x + 3h h) - 2x^ x - 3h i) 2x^ x2 + 3h j) 2x^ x2 - 3h k) - 2x^ x2 + 3h l) - 2x^ x2 - 3h m) 2x2 ^ x2 + 3h n) 2x2 ^ x2 - 3h o) - 2x2 ^ x2 + 3h p) - 2x2 ^ x2 - 3h q) 2x2 ^ x2 + 3xh r) 2x2 ^ x2 - 3xh s) - 2x2 ^ x2 + 3xh t) - 2x2 ^ x2 - 3xh Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 182 ♦ Chapter 5 - Polynomials West Point Grey Academy 3. Find each product. a) 3^ x - 4h b) - 3^ a - 4h c) 4^ y + 5h d) - 5^ b2 - 2h e) 3x^ x - 2h f) - 2y^ y - 1h g) - 2c^4c2 - ch h) xy^ x + yh i) m2 ^ m - 1h j) p2 ^- 2p + 4h k) - ab^ a2 + b2h l) - d 2 ^ d 2 - eh m) 3xy^- 2x + 5yh n) - 2n2 ^3n - 4h o) 2z2 ^ z2 - 3zh p) 4tp^- 2t2 + 3ph q) 2x2 ^- x2 + 3xh r) 2x2 ^- x2 - 3xh s) - 2x2 ^- x2 + 3xh t) - 2x2 ^- x2 - 3xh u) 2^ x2 - 3y + 4zh v) 2x^ x + 2y - 3zh w) 2x2 ^- x2 + 4y + 3zh x) - 2x2 ^3x2 - 2y2 + 4z2h Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.3 - Multiplying Polynomials ♦ 183 West Point Grey Academy 4. Determine the algebra tile multiplication indicated. a) b) c) d) Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 184 ♦ Chapter 5 - Polynomials West Point Grey Academy 5. Fill in the algebra tile operation a) b) c) d) e) Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.3 - Multiplying Polynomials ♦ 185 West Point Grey Academy 6. Determine the area. a) b) 3x 5x 3x B C c) d) 2x AC = 12x BD = 8x - 2 5x + 2 A D 7. Determine the volume. a) b) 2x x 2x 2x x c) d) x x 2x 4x Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 186 ♦ Chapter 5 - Polynomials West Point Grey Academy 8. Explain the error, then write the correct solution. a) - 2^ x - 3h =- 2x - 6 b) 5^2x - 4h = 10x - 15 c) - 2x^ x + 7h =- 2x2 - 14 d) 8^- 3x - 1h =- 24x + 8 9. Determine the surface area of the open top box. a) b) x x x x 10. Determine the surface area of the right triangular prism. a) b) 5x 4x 12x x 3x 11. Determine the surface area. a) x b) 2x + 1 2x + 3 2x Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.4 - Dividing Polynomials ♦ 187 West Point Grey Academy 5.4 Dividing Polynomials Dividing a Polynomial by a Constant To divide a polynomial by a constant, divide each term of the polynomial by the constant. a+b+g+z a b z c = c + c +g+ c , c ! 0 2 Example 1 Divide 9x - 3x + 6. 3 ►Solution: 9x2 - 3x + 6 9x2 3x 6 3x2 x 2 3 = 3 - 3 +3= - + 2 Example 2 Divide 16x - 12x + 8 -4 ►Solution: 16x 2 - 12x + 8 = 16x 2 - 12x + 8 =- 4x 2 + 3x - 2 -4 -4 -4 -4 Dividing a Polynomial by a Monomial To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. a+b+g+z a b z c = c + c +g+ c , c ! 0 9x2 + 6xy Example 3 Divide - 3x ►Solution: - 9x2 + 6xy 9x 2 6 xy 3x = - + = - 3x + 2y 3x 3x - 25x 2 y + 10xy 2 - 5xy Example 4 Divide - 5xy - 25x 2 y + 10xy 2 - 5xy - 25x 2 y 10xy 2 5xy ►Solution: = + - = 5x - 2y + 1 - 5xy - 5xy - 5xy - 5xy Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 188 ♦ Chapter 5 - Polynomials West Point Grey Academy Using Algebra Tiles for Division When dividing by a monomial with algebra tiles, the process of multiplication is reversed. The solution will be found on the left column of the table. 2 Example 5 Divide - 2x + 4x with algebra tiles. 2x 2x ►Solution: - 2x2 + 4x × = Therefore the unknown value is because x $ ^- xh =- x2. x - x2 -x = Therefore the unknown value is because x $ 2 = 2x. 2 x 2x Therefore the algebra tile diagram must have: 2 or - 2x + 4x =- x + 2 2x Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.4 - Dividing Polynomials ♦ 189 West Point Grey Academy 5.4 Exercise Set 1. Divide. a) 2x + 4 b) 2x - 4 2 2 c) - 2x + 4 d) - 2x - 4 2 2 6x + 12y 6x - 12y e) f) 3 3 6x 12y 6x 12y g) - + h) - - 3 3 i) - 10x + 25y j) 10x - 25y -5 -5 - 10x - 25y - 10x + 5 k) l) -5 5 2. Divide. 2 2 a) 5x - 10x - 15 b) 5x - 10x + 15 5 5 - 5x2 + 10x - 15 2 c) 5 d) - 5x - 10x - 15 5 2 e) 5x - 10x + 15 f) - 5x2 - 10x + 15 -5 -5 5x2 - 10xy + 15y2 5x2 + 10xy - 15y2 g) h) - -5 -5 i) - 5x 2 + 10x + 5 j) - 5x 2 + 10x - 5 -5 -5 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 190 ♦ Chapter 5 - Polynomials West Point Grey Academy 3. Divide. 2 2 a) 6x + x b) 6x - x x x 2 2 c) - 6x + x d) - 6x - x x x 2 e) - 6x + x f) 6x2 - x -x -x 9x2 + 15xy 9x2 - 15xy g) h) 3x 3x i) - 9x2 + 15xy j) - 9x2 - 15xy 3x 3x 9x2 + 15xy 9x2 + 15xy k) - l) - 3x - 3x 3 2 3 2 m) 9x - 6x + 3x n) - 9x + 6x - 3x 3x 3x 3 2 3 2 o) 9x - 6x - 3x p) - 9x - 6x + 3x 3x 3x 3 2 q) 9x - 6x + 3x r) - 9x 3 + 6x 2 - 3x - 3x - 3x s) 9x 3 - 6x 2 - 3x t) - 9x 3 - 6x 2 + 3x - 3x - 3x Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.4 - Dividing Polynomials ♦ 191 West Point Grey Academy 4. Use algebra tiles to perform each division. a) 2x2 - 4 2 b) - 3x 2 + 3x 3 c) - 2x2 + 4x 2x d) 3x2 - 6x 3x Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 192 ♦ Chapter 5 - Polynomials West Point Grey Academy 5. Determine the height. a) b) Area = 10x 2 - 15x Area = 12x 2 - 6x 5x 4x c) d) Volume = 18x - 27x 2 Volume = 24x 2 - 20x x 3 3x 6. The perimeter of a square is ^8x2 + 4x - 16h meters. 7. The area of a parallelogram is ^12x2 - 6xh square Find the length of each side. metres. If its base is 3x meters, what is its height? 8. The volume of a rectangular solid shaped swimming 9. The volume of a pyramid with a rectangular base is pool is ^96x 2 - 24x h square metres. If its width is ^12x 2 - 4x h square metres. If the width of the base 3x m, and its height is 2 m, determine its length. is 6 m and the length is ^3x - 1 h m, determine the height. Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.4 - Dividing Polynomials ♦ 193 West Point Grey Academy 10. Perform the combined operations. ^3x2 - 6x + 9h a) 2x^ x + 3h + 6x - 4x + 8 - 4x^2 - xh 2 b) - 2 3 - 2x2 ^3 - xh 6x^2x - 4h x^- 6x2 + 9xh 6 - 4x c) + d) - 2 2x 2 - 3x -^4x 2 - 2xh e) x^2x - 3y + 1h - - 3 ^1 - xh 4x2 + 8xy - 12x f) 4 2x ^ 8x 2 - 6x h 4x ^1 - xh ^6x - 9x 2 h ^8x 2 - 12x h g) + h) - 2x x 3x 4 - x ^6x 2 - 4x + 2 h ^6x - 4x 2 + 8 h ^2x - 3x 2 + 5 h i) - j) - 2x ^2 - 3x h 2x 2 -1 - 3x ^8x - 4x 2 + 6 h 4x ^6 - 8x h - 5x ^3 - 6x h 4x ^10x - 5 h k) + l) - -2 2 3 5 Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 194 ♦ Chapter 5 - Polynomials West Point Grey Academy 5.5 Chapter Review Section 5.1 1. Determine the degree of the polynomials. a) 5 3 b) 2 2 y 2 c) 3x 3 + 2x 2 y 2 d) 3 x 3 - 3x 2 + 5 2. Simplify each polynomial. a) 2x 2 - y 2 + 3x 2 + 2y 2 b) 4x 2 - 3y + 2x - y c) - 2xy 2 - x 2 y + xy 2 - x 2 y d) - 2xy - 3yx + 4xy + 5yx 3. Write a polynomial expression in simplified form for the algebra tiles. a) b) Section 5.2 4. Add. a) ^3 - 2x 2 + 4x h + ^- 2x + x 2 - 5 h b) ^2xy + x - y h + ^- xy - 2x - y h c) ^- 5 + 3x 2 + 7x h + ^- x 2 + 4 - 3x h d) ^- 4xy - x + 3y h + ^- xy + 7y - 3x h Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.  Section 5.5 - Chapter Review ♦ 195 West Point Grey Academy 5. Add using algebra tiles. a) ^2x - x 2 + 3 h + ^- 3x + 2x 2 - 1 h b) ^- 2xy + x + 2y h + ^- xy - x - yh 6. Subtract. a) ^- 3x + 2x 2 + 4h - ^- 2 - x 2 + 4x h b) ^2xy - x + 3y h - ^ y - x + xy h c) ^4x - 5x 2 - 1h - ^5 - 3x 2 - 2x h d) ^3x - 2x 2 + 4 h - ^- 2x 2 + 3x - 4 h Section 5.3 7. Find the product. a) 3 ^ x 2 - 2x + 3 h b) - 3x ^- x 2 + 2x - 3 h c) 3x 2 ^2x 2 - x + 1 h d) - 2xy ^- x 2 + 2xy + y 2 h e) 2xy 2 ^ x 2 - xy + y 2 h f) - 2xyz ^ x - y + z h g) 3xy ^2x - 3y + 2z h h) - 2xyz ^- x 2 + y 2 - z 2 h i) - 2xy ^3x - y + z h j) 3xyz ^- 2x 2 - 3y 2 + 4z 2 h k) 2x 2 y ^- x + 2y - 3z h l) - 3y 2 z ^- 2x + y - 3z h Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher. 196 ♦ Chapter 5 - Polynomials West Point Grey Academy Section 5.4 8. Find the quotient. a) 3x 2 - 9x + 6 2 b) - 4x + 8x + 6 3 -2 10x 2 - 5xy - 7x 2 + 21xy c) d) 5x - 7x - 4x 2 + 8xy - 12x a 2 bc - ab 2 c + abc 2 e) f) 2x - abc g) - a 2 b 2 c + ab 2 c 2 - a 2 b 2 c 2 2 2 2 h) - a bc - a bc + abc 2 ab 2 c - abc 9. Perform the combined operations. - 8x 2 ^2 - x h a) 3x ^2x - 3 h - 6x + 4x - 8 + 3x ^ 2 - x h 2 b) 2 4x - 6x 2 ^4 - 3x h c) - 2x ^3x - 2y + 1 h - - 2x ^3 - 2x h 8x - 4x 2 + 2xy d) 2 2x - 3 x 2 ^ 2 - 5x h - 4x 2 ^3x - 3y + 2h 6x 2 ^3 - 2x + y h e) + 3x ^4x - 3 h f) - -x 2x -x Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.