Question

complete the following questions and show all of your work in order to receive full credit.

1. which expression is equivalent to $2x^2 - 18$?

a. $2(x - 3)^2$
b. $(x + 9)(x - 9)$
c. $2(x + 3)(x - 3)$
d. $(2x + 6)(2x - 6)$

1. which expression is equivalent to $x^4 - 2x^3 + x^2$?

a. $x^2(x - 1)^2$
b. $x^2(x + 1)^2$
c. $x^2(x + 1)(x - 1)$
d. $x^2(x^2 + 1)(x - 1)$

1. which expression is equivalent to $x^2 - 64$?

a. $(x - 4)(x + 4)$
b. $(x - 8)(x + 8)$
c. $(x - 4)^2$
d. $(x - 8)^2$

1. which expression below shows $x^2 + 5x - 6$ in factored form?

a. $(x + 1)(x - 6)$
b. $(x + 3)(x + 2)$
c. $(x + 6)(x - 1)$
d. $(x + 2)(x - 3)$

1. which expression is equivalent to $12x^2 + x - 20$?

a. $(3x + 4)(4x - 5)$
b. $(3x - 4)(4x + 5)$
c. $(4x + 3)(5x - 4)$
d. $(4x - 3)(5x + 4)$

Explanation:

Question 1

Step1: Factor out the common term

First, factor out the greatest common factor (GCF) from \(2x^{2}-18\). The GCF of \(2x^{2}\) and \(18\) is \(2\), so we get \(2(x^{2} - 9)\).

Step2: Apply the difference of squares

The expression \(x^{2}-9\) is a difference of squares, which can be factored as \((x + 3)(x - 3)\) (since \(a^{2}-b^{2}=(a + b)(a - b)\) with \(a = x\) and \(b = 3\)). So, \(2(x^{2}-9)=2(x + 3)(x - 3)\).

Step1: Factor out the common term

Factor out the greatest common factor from \(x^{4}-2x^{3}+x^{2}\). The GCF of \(x^{4}\), \(-2x^{3}\), and \(x^{2}\) is \(x^{2}\), so we have \(x^{2}(x^{2}-2x + 1)\).

Step2: Factor the quadratic

The quadratic \(x^{2}-2x + 1\) is a perfect square trinomial, which factors as \((x - 1)^{2}\) (since \((a - b)^{2}=a^{2}-2ab + b^{2}\) with \(a = x\) and \(b = 1\)). Thus, \(x^{2}(x^{2}-2x + 1)=x^{2}(x - 1)^{2}\).

Step1: Identify the form

The expression \(x^{2}-64\) is a difference of squares, where \(a = x\) and \(b = 8\) (since \(64 = 8^{2}\)).

Step2: Apply the difference of squares formula

Using the formula \(a^{2}-b^{2}=(a - b)(a + b)\), we get \(x^{2}-64=(x - 8)(x + 8)\).

Answer:

c. \(2(x + 3)(x - 3)\)

Question 2