4.11 Chapter 4 Example Solutions

Example 1: Factoring the Greatest Common Factor

1. First, find the GCF of the expression. The GCF of 6, 45, and 21 is 3. The GCF of [latex]x^3[/latex], [latex]x^2[/latex], and [latex]x[/latex] is [latex]x[/latex]. (Note that the GCF of a set of expressions in the form [latex]x^n[/latex] will always be the exponent of lowest degree.) And the GCF of [latex]y^3[/latex], [latex]y^2[/latex], and [latex]y[/latex] is [latex]y[/latex]. Combine these to find the GCF of the polynomial, [latex]3xy[/latex].

Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]3xy\left(2x^2y^2\right)=6x^3y^3[/latex], [latex]3xy\left(15xy\right)=45x^2y^2[/latex], and [latex]3xy\left(7\right)=21xy[/latex].

Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.

[latex]\left(3xy\right)\left(2x^2y^2+15xy+7\right)[/latex]

2. [latex]\left(b^2-a\right)\left(x+6\right)[/latex]

Example 1: Factoring a Trinomial with Leading Coefficient 1

1. We have a trinomial with leading coefficient [latex]1[/latex], [latex]b=2[/latex], and [latex]c=−15[/latex]. We need to find two numbers with a product of [latex]−15[/latex] and a sum of [latex]2[/latex]. In Table 1, we list factors until we find a pair with the desired sum.

Now that we have identified [latex]p[/latex] and [latex]q[/latex] as [latex]−3[/latex] and [latex]5[/latex], write the factored form as [latex]\left(x-3\right)\left(x+5\right)[/latex].

2. [latex]\left(x−6\right)\left(x−1\right)[/latex]

Example 1: Factoring a Trinomial by Grouping

1. We have a trinomial with [latex]a=5[/latex], [latex]b=7[/latex], and [latex]c=−6[/latex]. First, determine [latex]ac=−30[/latex]. We need to find two numbers with a product of [latex]−30[/latex] and a sum of [latex]7[/latex]. In Table 2, we list factors until we find a pair with the desired sum.

So [latex]p=−3[/latex] and [latex]q=10[/latex].

$\begin{array}{cc}5x^2-3x+10x-6\hfill & \text{Rewrite the original expression as}a{x}^2+px+qx+c.\hfill \\ x(5x-3)+2(5x-3)\hfill & \text{Factor out the GCF of each part}.\hfill \\ (5x-3)(x+2)\hfill & \text{Factor out the GCF}\text{}\text{of the expression}.\hfill \end{array}$ 

2. a. [latex]\left(2x+3\right)\left(x+3\right)[/latex]

b. [latex]\left(3x−1\right)\left(2x+1\right)[/latex]

Example 1: Factoring a Perfect Square Trinomial

1. Notice that [latex]25x^2[/latex] and [latex]4[/latex] are perfect squares because [latex]25x^2 = (5x)^2[/latex] and [latex]4 = 2^2[/latex]. Then check to see if the middle term is twice the product of [latex]5x[/latex] and [latex]2[/latex]. The middle term is, indeed, twice the product: [latex]2(5x)(2) = 20x[/latex]. Therefore, the trinomial is a perfect square trinomial and can be written as [latex](5x+2)^2[/latex].

2. [latex]\left(7x-1\right)^2[/latex]

Example 1: Factoring a Difference of Squares

1. Notice that [latex]9x^2[/latex] and [latex]25[/latex] are perfect squares because [latex]9x^2 = (3x)^2[/latex] and [latex]25 = 5^2[/latex]. The polynomial represents a difference of squares and can be written as [latex](3x + 5)(3x - 5)[/latex].

2. [latex]\left(9y-10\right)\left(9y-10\right)[/latex]

Example 1: Factoring a Sum of Cubes

1. Notice that [latex]x^3[/latex] and [latex]512[/latex] are cubes because [latex]8^3 = 512[/latex]. Rewrite the sum of cubes as [latex](x + 8)(x^2 - 8x +64)[/latex].

2. [latex]\left(6a+b\right)\left(36a^2-6ab+b^2\right)[/latex]

Example 2: Factoring a Difference of Cubes

1. Notice that [latex]8x^3[/latex] and [latex]125[/latex] are cubes because [latex]8x^3 = (2x)^3[/latex] and [latex]125 = 5^3[/latex]. Write the difference of cubes as [latex](2x - 5)(4x^2 + 10x + 25)[/latex].

2. [latex]\left(10x-1\right)\left(100x^2-10x+1\right)[/latex]

Example 1: Factoring Expressions with Fractional or Negative Exponents

1. Factor out the term with the lowest value of the exponent. In this case, that would be [latex]{(x\;+\;2)}^\frac{-1}3[/latex].

[latex]{(x\;+\;2)}^\frac{-1}3(3x\;+\;4(x\;+2))\;\;\;\;\;\;\;\;\;\;\;\;\;Factor\;out\;the\;GCF.\\{(x\:+\;2)}^\frac{-1}3(3x\;+4x\;+8)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Simplify.\\{(x\;+\;2)}^\frac{-1}3(7x\;+8)[/latex]

2. [latex]\left(5a-1\right)^\frac{-1}{4}\left(17a-2\right)[/latex]