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

### Guidance

In this Concept, we will learn how to factor quadratic polynomials for different values of

**Factoring Quadratic Expressions in Standard Form**

**Quadratic polynomials** are polynomials of degree 2. The standard form of a quadratic polynomial is *where* *and* *are real numbers*.

#### Example A

*Factor*

**Solution:** 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

So the answer is

We can check to see if this is correct by multiplying

2 is multiplied by

Combine the like terms:

#### Example B

*Factor*

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

The number 8 can be written as the product of the following numbers.

*And*

The answer is

#### Example C

*Factor*

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

In this case, we must take the negative sign into account. The number –15 can be written as the product of the following numbers.

**And also,**

*and* *This is the correct choice.*

*and*

The answer is

**To Summarize:**

A quadratic of the form

- 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 both \begin{align*}m\end{align*} and \begin{align*}n\end{align*}are negative.- Example: \begin{align*}x^2-6x+8\end{align*} factors as \begin{align*}(x-2)(x-4)\end{align*}.

- If \begin{align*}c\end{align*} is negative then either \begin{align*}m\end{align*} is positive and \begin{align*}n\end{align*}is negative or vice-versa.
- Example: \begin{align*}x^2+2x-15\end{align*} factors as \begin{align*}(x+5)(x-3)\end{align*}.
- Example: \begin{align*}x^2+34x-35\end{align*} factors as \begin{align*}(x+35)(x-1)\end{align*}.

- If \begin{align*}a=-1\end{align*}, factor a common factor of –1 from each term in the trinomial and then factor as usual. The answer will have the form \begin{align*}-(x+m)(x+n)\end{align*}.
- Example: \begin{align*}-x^2+x+6\end{align*} factors as \begin{align*}-(x-3)(x+2)\end{align*}.

### Video Review

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### Guided Practice

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

**Solution:** 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+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 \qquad This \ is \ the \ correct \ choice.\end{align*}

The answer is \begin{align*}-(x-3)(x+2)\end{align*}.

### Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Factoring Quadratic Equations (16:30)

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*}.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.8.