Algebra Class: Variables and Constants
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Questions and Answers

Which of the following options correctly identifies the degree of the polynomial 13x⁴ + 4y⁷x?

  • 8
  • 4 (correct)
  • 11 (correct)
  • 7

The expression 2xy + 5y + 8 is a trinomial.

True (A)

What is the coefficient of xy in the expression 4y12xy + 32?

48

The expression 3x + 4y + 2xy + 5x - xy - 2x simplifies to ___

<p>6x + 3y + xy</p> Signup and view all the answers

Match the following expressions with their classifications.

<p>14abc + b = Binomial 3a5ab + 2c = Monomial 18pq = Monomial 5x + y = Binomial 2x - 5yz + 3z = Trinomial</p> Signup and view all the answers

What is the result of multiplying 3ab by 2a?

<p>6a²b (D)</p> Signup and view all the answers

The expansion of (a + 3)(a + 5) results in a² + 8a + 15.

<p>True (A)</p> Signup and view all the answers

What is the product of (2x + 1)(x + 4)?

<p>2x² + 9x + 4</p> Signup and view all the answers

The result of expanding (3x + 1)(2x + 5y + 6) is _____

<p>6x² + 15xy + 18x + 2x + 6</p> Signup and view all the answers

Match the expressions with their respective products:

<p>(3x - 1)(x - 5) = 2x² - 8x + 5 (2a - 3)(3a + 4) = 6a² - 1 (4y - 1)(5y + 3) = 20y² - 27y - 3 (2z + 1)(z - 7) = 2z² - 13z - 7</p> Signup and view all the answers

What is the expression for the distance travelled by the car in terms of x?

<p>(70 - 3x)(x + 2) (D)</p> Signup and view all the answers

The area of the rectangular field is calculated by multiplying its length and width.

<p>True (A)</p> Signup and view all the answers

If $x = 3$, what is the distance travelled by the car?

<p>189 km</p> Signup and view all the answers

The area of the rectangular field is ________ m² in terms of x.

<p>(3x - 2)(x + 5)</p> Signup and view all the answers

Match the following expressions with their operations:

<p>(70 - 3x)(x + 2) = Distance travelled (3x - 2)(x + 5) = Area of rectangle (2x - 5)(3x + 7) = Coefficient of x x + 2 = Time taken by the car</p> Signup and view all the answers

What is the result of dividing $12x^4$ by $3x^2$?

<p>$4x^2$ (B)</p> Signup and view all the answers

Dividing $39m^5$ by $13m^3$ results in $3m^2$.

<p>True (A)</p> Signup and view all the answers

What is the remainder when dividing $7x^3 + 13x^2 - 16x + 23$ by $7x - 1$?

<p>4</p> Signup and view all the answers

The result of dividing $18x^2 + 12x$ by $6x$ is _____ .

<p>3x + 2</p> Signup and view all the answers

Match the following polynomials with their corresponding divisor:

<p>$2x^3 + 19x^2 + 6x + 63$ = $x + 4$ $x^3 - 7x^2 + 10x$ = $x - 2$ $6x^6 - 3x^2 + 7x - 152$ = $x^2 - 3$ $x^4 + 4x^3 + 6x^2 + 2$ = $x + 1$</p> Signup and view all the answers

What is the result of expanding (3m + 5n)²?

<p>9m² + 30mn + 25n² (A)</p> Signup and view all the answers

(a + b)² = a² + 2ab + b² is a standard identity.

<p>True (A)</p> Signup and view all the answers

What do you call a polynomial with two terms?

<p>binomial</p> Signup and view all the answers

The expression (x + 2)(x - 2) simplifies to __________.

<p>x² - 4</p> Signup and view all the answers

Match each polynomial type with its definition:

<p>Monomial = A polynomial with one term Binomial = A polynomial with two terms Trinomial = A polynomial with three terms Polynomial = An algebraic expression with one or more terms</p> Signup and view all the answers

Study Notes

Variables and Constants

  • Variables are symbols representing unknown values, such as a, b, c, x, y, z, p.
  • Constants are fixed values in algebraic expressions, e.g., 8, 10, 5, 6.

Types of Algebraic Expressions

  • Monomials: Expressions with one term, e.g., 3a5ab.
  • Binomials: Expressions with two terms, e.g., 14abc + b.
  • Trinomials: Expressions with three terms, e.g., 2x - 5yz + 3z.

Coefficients

  • The coefficient is the numerical factor in a term, e.g., in 9xy, the coefficient is 9.

Degree of Polynomials

  • The degree of a polynomial indicates the highest power of the variable, e.g., in 256z² + 8, the degree is 2.

Addition and Subtraction of Expressions

  • Algebraic expressions can be combined through addition and subtraction by collecting like terms, e.g., 11a² + 2ab + 8 and 4ab + 6 simplifies to 11a² + 6ab + 14.

Simplification Techniques

  • Collecting like terms helps simplify expressions, e.g., 3x + 4y + 2xy + 5x - xy - 2x reduces to 6x + 3y + xy.

Multiplication of Algebraic Expressions

  • Use the distributive property to multiply terms, e.g., (a + b)(c + d) = ac + ad + bc + bd.

Important Algebraic Identities

  • The square of a binomial: (a + b)² = a² + 2ab + b².
  • The difference of squares: (a + b)(a - b) = a² - b².

Division of Polynomials

  • Dividing monomials involves subtracting exponents, e.g., x^m ÷ x^n = x^(m-n).
  • Polynomials can be divided by linear expressions, leading to a quotient and possibly a remainder.

Problem Solving Techniques

  • For problem solving, setting up an expression based on given conditions helps in finding distances, areas, and solutions.

Comparative Areas

  • To compare areas of shapes (e.g., square vs. rectangular window), calculate area using length and width and subtract to find the difference.

Overview of Polynomials

  • A polynomial is an algebraic expression with non-negative integer exponents, categorized into monomials, binomials, and trinomials based on the number of terms.

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Description

This quiz focuses on identifying variables and constants in various algebraic expressions. Participants will analyze different expressions and classify the components accurately. It serves as a valuable exercise for students in algebra to enhance their understanding of these fundamental concepts.

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