FNDMATH Module 1 Handout - Basic Algebra PDF

Lesson 1 FNDMATH Lecture 1 Algebraic Expressions Integer Exponents Addition/Subtraction of Algebraic Expressions Multiplicatio...

Lesson 1 FNDMATH Lecture 1 Algebraic Expressions Integer Exponents Addition/Subtraction of Algebraic Expressions Multiplication/Division of Algebraic Expressions 1 1 CONSTANTS AND VARIABLES CONSTANT Any symbol whether it be a number or a letter which is used to represent a fixed value VARIABLE A single letter may represent a set of real numbers the values of which depend upon the situation or problem being analyzed The value of the letter is not fixed 2 FNDMATH 1 Lesson 1 ALGEBRAIC EXPRESSIONS An expression involving constants/or variables with all or some of the four algebraic operation of addition, subtraction, multiplication and division example 2 xy 2 x3 Monomial 3 ALGEBRAIC EXPRESSIONS TERM Each monomial in an expression BINOMIAL An expression is composed only of two terms MULTINOMIAL An expression is composed of more than two terms 4 FNDMATH 2 Lesson 1 DEGREE OF MULTINOMIAL DEGREE OF A TERM With respect to a literal factor is the exponent of that literal factor in the term 3x2 y3z x = 2nd degree y = 3rd degree z = 1st degree 5 POWERS AND EXPONENTS 2 x3 y2 2  x3  y2 Factors of expressions POWER Composed of a base and an exponent EXPONENT Small number written at the upper right hand corner of the base It indicates how many times the base occurs as a factor 6 FNDMATH 3 Lesson 1 POWERS AND EXPONENTS For every b , n  R (n, a positive integer) b n  b  b  b  b  b  b...b bn is read as “nth power of b” or “b raised to the nth power” x 2 x y  x yz   2xxx y yz  2 x3 y2z 7 RULES FOR POSITIVE INTEGRAL EXPONENTS 1. For every a , m , n  R (m, n a positive integer) a m  a n  a mn 1. 2 x 2 y   3 x y 4 2. 4 w 2 z 3  2 wz  5 w3 z 2  8 FNDMATH 4 Lesson 1 RULES FOR POSITIVE INTEGRAL EXPONENTS 2. For every a,m,n  R (m > n ) (a ≠ 0) am n  a mn a 27 x5z6w 7 1. 12 x2z 4w 9 RULES FOR POSITIVE INTEGRAL EXPONENTS 3. For every a,b,m  R (m> 0 ) a b  m  a mbm 1. 3 a b c  3 10 FNDMATH 5 Lesson 1 RULES FOR POSITIVE INTEGRAL EXPONENTS 4. For every a,b,m  R (m> 0 ); (b ≠ 0) m  a  am    m  b  b 7  x  1.   yz  11 RULES FOR POSITIVE INTEGRAL EXPONENTS 5. For every a,m,n  R (m,n > 0 ) a  n m  a mn 1.  2 x 2 y 3 z  3 12 FNDMATH 6 Lesson 1 EXAMPLES Simplify the following expressions using the rules on positive integer exponents. All exponents in the answers must be positive 1. 25 x2 y 24 x5 y2 15 x2 y 4 6. 25x3 y  3  r s 2t 4   3  r 4 s t 5 4 4 2. 3w 2v 2 7. 3. 4 3 n 1  8 5 n 1  1 2 w 5v 7 2 a n 6 a m   3 2  2 2 4. x3 y5 8. 16 x5 y3 m 5. x  x 5m  x 2m  3 a b c   4 a b c  2 4 2 3 9. 2 4 a b c  2 5 2 3 13  3  rs 2t 4   3  r 4 s t 5 4 3 n 1  8 5 n 1 4 4 2. 3. 15x2 y4 6 a  2 4. 2a n m 6. 25 x3 y 14 14 FNDMATH 7 Lesson 1  3 2  2 x3y5 8. 16 x5 y3  3 a b c   4 a b c  2 4 2 3 9. 2 4 a b c  2 5 2 3 15 15 ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS MONOMIALS Addition and subtraction of monomials can be done only if the monomials to be added are similar. To add or subtract similar terms, use the reverse of the distributive rule to factor out the common literal parts and then add or subtract their numerical parts Example: (ADD) 1. x2y  3x2y  9x2 y 5x2 y5z  3x2 y5z  7 x2 y5z 16 FNDMATH 8 Lesson 1 ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS MONOMIALS Example: (SUBTRACT) 3.  1 0 a 2b 4. 4x2 y  2 y 5x  x  3  6x  2x2    3a 2b 17 ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS MULTINOMIAL If the multinomial to be added or to be subtracted do not contain any similar terms, no combination of terms among the multinomial can be done. Combination is made easier if similar terms are arranged first, and then, the terms in each column are added or subtracted 18 FNDMATH 9 Lesson 1 ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS MULTINOMIAL Example: (ADD) 2 1. 3 x y  2 x 2 y  x y 2  3;  2 x y  5 xy  6 x  4 x 2 y; 7  2 xy 2  4 x  2 xy Arrange in descending power of x 2 x 2 y  3 xy 2  2 x  4 19 MULTIPLICATION OF ALGEBRAIC EXPRESSIONS MULTINOMIAL BY A MONOMIAL If the multinomial is to be multiplied by a monomial, the distributive rule is used. The product is the sum of all the products formed by multiplying each term of the multinomial by monomial multiplier EXAMPLE: 1. 2 x y 2 3 x 2  2 y  5  20 FNDMATH 10 Lesson 1 MULTIPLICATION OF ALGEBRAIC EXPRESSIONS MULTINOMIAL BY A MULTINOMIAL The distributive rule is once more used to multiply two multinomial EXAMPLE: 1. 2 x  3y  x 2  3 xy  21 MULTIPLICATION OF ALGEBRAIC EXPRESSIONS MULTINOMIAL BY MULTINOMIAL EXAMPLE: 2.  3 m 3  m 2  2 m  5  m 2  5m  6 3m 3  m 2  2m  5 m 2  5m  6 3m 5  m 4  2m 3  5m 2 15m 4  5m 3  10m 2  25m 18m 3  6m 2  12m  30 3m 5  14m 4  25m 3  m 2  37m  30 22 FNDMATH 11 Lesson 1 DIVISION OF ALGEBRAIC EXPRESSIONS MONOMIAL BY A MONOMIAL Divide a monomial by another monomial use the fundamental theorem fractions or rules of exponent. Zero exponent For each a  R a  0 a0  1 EXAMPLE: 16 x2 y 2 2  8 x 22 1.  24 x2 y5 3  8 y 52 2x0 2  3 or 3y 3y3 23 DIVISION OF ALGEBRAIC EXPRESSIONS MUTLINOMIAL BY A MONOMIAL To divide a multinomial by a monomial use the distributive rule and rules on exponent EXAMPLE: 1 1. 6 x 2 y 3  12 xy  4 xy 2  3 xy 6 x 2 y 3  12 xy  4 xy 2  3 xy 6x2y3 12 xy 4 xy 2    3 xy 3 xy 3 xy 4  2 xy 2  4  y 3 24 FNDMATH 12 Lesson 1 DIVISION OF ALGEBRAIC EXPRESSIONS MUTLINOMIAL BY A MONOMIAL Step 1: Arrange the terms of the multinomial according to descending powers of a literal factor common to both dividend and divisor Step 2: Divide the first term of the dividend by the first term of the divisor by applying rules on exponent during the process and proceeding the arithmetic division Step 3: Repeat step 2 until the remainder is of a lower degree(with respect to the chosen literal factor) than of the divisor 25 DIVISION OF ALGEBRAIC EXPRESSIONS MUTLINOMIAL BY A MULTINOMIAL EXAMPLE: 1. 4 x 3 y  8 x 2 y 2  x y 3  y 4 d iv id e b y 2x  y 26 FNDMATH 13 Lesson 1 EXERCISES: Perform the indicated operations and simplify 2 y  2x2 y  4x2 y 7. Divide 1. Add: x 2. Subtract: 1 5 w v from 1 2 w v  11x 2  y  5 x 3  3 y 3  10 xy 2 by 3. Add: 3 x  2 y  4 z 4 y  2 x  using powers of y as basis* 2x  3y  3z 8. 1 6 x 3  4 x 2  1 9 4x  5 y  7 z d iv id e by 4. Subtract: 5 x 3  7x  3 from 4x  5 3x  x 2  5 5. ( 2 x 2 y 3 ) (  3 y z 2 ) 6. 4 x 2 y 3 ( 2 x y 3  3 x z 2 ) 27 1. x2 y  2x2 y  4x2 y 2. Subtract: 5x3  7 x  3 from 3x  x2  5 3. 4 x 2 y 3 (2 xy 3  3 xz 2 ) 4. 16 x3  4 x2  19 d iv id e by 4x  5 28 28 FNDMATH 14 Lesson 1 4. 16 x3  4 x2  19 d iv id e by 4x  5 29 29 Challenging problem… 9 m 3n  3 2 m 3  42m 6  15n 2  6  n d iv id e by 7m 3  5n  3 30 30 FNDMATH 15