# 4.11 Chapter 4 Example Solutions

## 4.2 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 $x^3$, $x^2$, and $x$ is $x$. (Note that the GCF of a set of expressions in the form $x^n$ will always be the exponent of lowest degree.) And the GCF of $y^3$, $y^2$, and $y$ is $y$. Combine these to find the GCF of the polynomial, $3xy$.

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

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

$\left(3xy\right)\left(2x^2y^2+15xy+7\right)$

2. $\left(b^2-a\right)\left(x+6\right)$

## 4.3 Example Solutions

### Example 1: Factoring a Trinomial with Leading Coefficient 1

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

Table 1
Factors of -15 Sum of Factors
$1, -15$ $-14$
$-1, 15$ $14$
$3, -5$ $-2$
$-3, 5$ $2$

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

2. $\left(x−6\right)\left(x−1\right)$

## 4.4 Example Solutions

### Example 1: Factoring a Trinomial by Grouping

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

Table 2
Factors of -30 Sum of Factors
1, -30 -29
-1, 30 29
2, -15 -13
-2, 15 13
3, -10 -7
-3, 10 7

So $p=−3$ and $q=10$.

2. a. $\left(2x+3\right)\left(x+3\right)$

b. $\left(3x−1\right)\left(2x+1\right)$

## 4.5 Example Solutions

### Example 1: Factoring a Perfect Square Trinomial

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

2. $\left(7x-1\right)^2$

## 4.6 Example Solutions

### Example 1: Factoring a Difference of Squares

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

2. $\left(9y-10\right)\left(9y-10\right)$

## 4.7 Example Solutions

### Example 1: Factoring a Sum of Cubes

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

2. $\left(6a+b\right)\left(36a^2-6ab+b^2\right)$

### Example 2: Factoring a Difference of Cubes

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

2. $\left(10x-1\right)\left(100x^2-10x+1\right)$

## 4.8 Example Solutions

### 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 ${(x\;+\;2)}^\frac{-1}3$.

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

2. $\left(5a-1\right)^\frac{-1}{4}\left(17a-2\right)$