Suppose your height above sea level in feet when traveling over a hill can be represented by the expression

### Factoring Quadratic Expressions

Previously, we factored common monomials, so you already know how to factor quadratic polynomials where

**Quadratic polynomials** are polynomials of degree 2. The standard form of a quadratic polynomial is

A quadratic of the form

There are some important rules to know when factoring quadratics:

- If
b andc are positive then bothm andn are positive.- Example:
x2+8x+12 factors as(x+6)(x+2) .

- Example:
- If
b is negative andc is positive then bothm andn are negative.- Example:
x2−6x+8 factors as(x−2)(x−4) .

- Example:
- If
c is negative then eitherm is positive andn is negative or vice-versa.- Example:
x2+2x−15 factors as(x+5)(x−3) . - Example:
x2+34x−35 factors as(x+35)(x−1) .

- Example:
- If
a=−1 , factor a common factor of –1 from each term in the trinomial and then factor as usual. The answer will have the form−(x+m)(x+n) .- Example:
−x2+x+6 factors as−(x−3)(x+2) .

- Example:

#### Let's factor the following polynomials:

x2+5x+6

We are looking for an answer that is the product of the two binomials in parentheses:

To fill in the blanks, we want two numbers

Since both

The number six can be written as the product of 1 and 6 or 2 and 3. Testing the product and sums we get:

So the answer is

We can check to see if this is correct by multiplying

2 is multiplied by

Combine the like terms:

- \begin{align*}x^2-6x+8\end{align*}

We are looking for an answer that is the product of the two binomials in parentheses: \begin{align*}(x+ \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )(x + \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )\end{align*}.

Since \begin{align*}b\end{align*} is negative and \begin{align*}c\end{align*} is positive, both of the blanks are going to be negative. We only want the negative factors of 8.

The number 8 can be written as the product of -1 and -8 or -2 and -4. Testing the product and sums, we get:

\begin{align*}-x^2+16x+63 = -(x^2-16x-63)\end{align*}

-2 and -4 are the factors that multiply to 8 and add to -6. This is the correct choice.

The answer is \begin{align*}(x-2)(x-4)\end{align*}.

- \begin{align*}x^2+2x-15\end{align*}

We are looking for an answer that is the product of the two binomials in parentheses: \begin{align*}(x \pm \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )(x \pm \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )\end{align*}.

Since \begin{align*}b\end{align*} is positive and \begin{align*}c\end{align*} is negative, one of the blanks will be negative and one will be positive. The number –15 can be written as the product of -1 and 15, 1 and -15, 3 and -5, or -3 and 5. Testing the product and sums, we get:

\begin{align*}-15 = -1 \cdot 15 && and && -1 + 15 = 14\\ -15 = 1 \cdot -15 && and && 1 + (-15) = -14\\ -15 = -3 \cdot 5 && and && (-3) + 5 = 2\\ -15 = 3 \cdot -5 && and && 3 + -5 = -2\\\end{align*} -3 and 5 are the factors that multiply to -15 and add to 2. The answer is \begin{align*}(x-3)(x+5)\end{align*}.

### Examples

#### Example 1

Earlier, you were told that your height above sea level when traveling over a hill can be represented by the expression \begin{align*}-x^2 + 18x + 63\end{align*}, where \begin{align*}x\end{align*} is the horizontal distance traveled. What is the factored form of this expression?

First, factor the common factor of -1 from each term in the trinomial. Factoring -1 changes the signs of each term in the expression.

\begin{align*}-x^2+18x+63 = -(x^2-18x-63)\end{align*}

We are looking for an answer that is the product of the two binomials in parentheses: \begin{align*}(x \pm \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )(x \pm \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )\end{align*}.

Now our job is to factor \begin{align*}x^2-18x-63\end{align*}. Since \begin{align*}b\end{align*} is negative and \begin{align*}c\end{align*} is negative, one of the blanks is positive and one is negative.

The number –63 can be written as the product of -1 and 63, 1 and -63, -3 and 21, 3 and -21, -7 and 9, or 7 and -9. Testing the products and sums, we get:

\begin{align*}&-63=(-1) \times 63 \qquad and \qquad (-1)+63=62\\ &-63=1 \times (-63) \qquad and \qquad 1+(-63)=-62\\ &-63=(-3) \times 21 \qquad and \qquad (-3)+21=18\\ &-63=3 \times (-21) \qquad and \qquad 3+(-21)=-18\\ &-63=(-7) \times 9 \qquad and \qquad (-7)+9=2\\ &-63=7 \times (-9) \qquad and \qquad 7+(-9)=-2\\ \end{align*}

3 and -21 are the factors that multiply to -63 and sum to -18.The answer is \begin{align*}-(x-21)(x+3)\end{align*}.

#### Example 2

Factor \begin{align*} -x^2+x+6\end{align*}.

Like in Example 1, first factor the common factor of –1 from each term in the trinomial.

\begin{align*}-x^2+x+6 = -(x^2-x-6)\end{align*}

We are looking for an answer that is the product of the two binomials in parentheses: \begin{align*}(x \pm \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )(x \pm \underline{\;\;\;\;\;\;\;\;\;\;\;} \ )\end{align*}.

Now our job is to factor \begin{align*}x^2-x-6\end{align*}.

The number –6 can be written as the product of the following numbers.

\begin{align*}&-6=(-1) \times 6 \qquad and \qquad (-1)+6=5\\ &-6=1 \times (-6) \qquad and \qquad 1+(-6)=-5\\ &-6=(-2) \times 3 \qquad and \qquad (-2)+3=1\\ &-6=2 \times (-3) \qquad and \qquad 2+(-3)=-1 \end{align*}

-3 and 2 are the factors that multiply to -6 and add to -1. The answer is \begin{align*}-(x-3)(x+2)\end{align*}.

### Review

Factor the following quadratic polynomials.

- \begin{align*}x^2+10x+9\end{align*}
- \begin{align*}x^2+15x+50\end{align*}
- \begin{align*}x^2+10x+21\end{align*}
- \begin{align*}x^2+16x+48\end{align*}
- \begin{align*}x^2-11x+24\end{align*}
- \begin{align*}x^2-13x+42\end{align*}
- \begin{align*}x^2-14x+33\end{align*}
- \begin{align*}x^2-9x+20\end{align*}
- \begin{align*}x^2+5x-14\end{align*}
- \begin{align*}x^2+6x-27\end{align*}
- \begin{align*}x^2+7x-78\end{align*}
- \begin{align*}x^2+4x-32\end{align*}
- \begin{align*}x^2-12x-45\end{align*}
- \begin{align*}x^2-5x-50\end{align*}
- \begin{align*}x^2-3x-40\end{align*}
- \begin{align*}x^2-x-56\end{align*}
- \begin{align*}-x^2-2x-1\end{align*}
- \begin{align*}-x^2-5x+24\end{align*}
- \begin{align*}-x^2+18x-72\end{align*}
- \begin{align*}-x^2+25x-150\end{align*}
- \begin{align*}x^2+21x+108\end{align*}
- \begin{align*}-x^2+11x-30\end{align*}
- \begin{align*}x^2+12x-64\end{align*}
- \begin{align*}x^2-17x-60\end{align*}

**Mixed Review**

- Evaluate \begin{align*}f(2)\end{align*} when \begin{align*}f(x)=\frac{1}{2} x^2-6x+4\end{align*}.
- Simplify \begin{align*}\sqrt{405}\end{align*}.
- Graph the following on a number line: \begin{align*}-\pi, \sqrt{2}, \frac{5}{3}, - \frac{3}{10}, \sqrt{16}\end{align*}.
- What is the multiplicative inverse of \begin{align*}\frac{9}{4}\end{align*}?

#### Quick Quiz

- Name the following polynomial. State its degree and leading coefficient: \begin{align*}6x^2 y^4 z+6x^6-2y^5+11xyz^4\end{align*}.
- Simplify \begin{align*}(a^2 b^2 c+11abc^5 )+(4abc^5-3a^2 b^2 c+9abc)\end{align*}.
- A rectangular solid has dimensions \begin{align*}(a+2)\end{align*} by \begin{align*}(a+4)\end{align*} by \begin{align*}(3a)\end{align*}. Find its volume.
- Simplify \begin{align*}-3hjk^3 (h^2 j^4 k+6hk^2)\end{align*}.
- Find the solutions to \begin{align*}(x-3)(x+4)(2x-1)=0\end{align*}.
- Multiply \begin{align*}(a-9b)(a+9b)\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.8.