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# Factorization of Quadratic Expressions

## Factor quadratics with positive coefficients

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Factorization of Quadratic Expressions

### Factorization of Quadratic Expressions

Quadratic polynomials are polynomials of the 2nd\begin{align*}2^{nd}\end{align*} degree. The standard form of a quadratic polynomial is written as

ax2+bx+c\begin{align*}ax^2 + bx + c\end{align*}

where a,b,\begin{align*}a, b,\end{align*} and c\begin{align*}c\end{align*} stand for constant numbers. Factoring these polynomials depends on the values of these constants. In this section we’ll learn how to factor quadratic polynomials for different values of a,b,\begin{align*}a, b,\end{align*} and c\begin{align*}c\end{align*}. (When none of the coefficients are zero, these expressions are also called quadratic trinomials, since they are polynomials with three terms.)

You’ve already learned how to factor quadratic polynomials where c=0\begin{align*}c = 0\end{align*}. For example, for the quadratic ax2+bx\begin{align*}ax^2 + bx\end{align*}, the common factor is x\begin{align*}x\end{align*} and this expression is factored as x(ax+b)\begin{align*}x(ax + b)\end{align*}. Now we’ll see how to factor quadratics where c\begin{align*}c\end{align*} is nonzero.

Factor when a = 1, b is Positive, and c is Positive

First, let’s consider the case where a=1,b\begin{align*}a = 1, b\end{align*} is positive and c\begin{align*}c\end{align*} is positive. The quadratic trinomials will take the form

x2+bx+c\begin{align*}x^2 + bx +c\end{align*}

You know from multiplying binomials that when you multiply two factors (x+m)(x+n)\begin{align*}(x + m)(x + n)\end{align*}, you get a quadratic polynomial. Let’s look at this process in more detail. First we use distribution:

(x+m)(x+n)=x2+nx+mx+mn\begin{align*}(x + m)(x + n) = x^2 + nx + mx + mn\end{align*}

Then we simplify by combining the like terms in the middle. We get:

(x+m)(x+n)=x2+(n+m)x+mn\begin{align*}(x + m)(x + n) = x^2 + (n + m) x +mn\end{align*}

So to factor a quadratic, we just need to do this process in reverse.

We see that x2+(n+m)x+mnis the same form asx2+bx+c\begin{align*}& \text{We see that} \qquad \qquad \qquad \quad \ x^2 + (n + m)x + mn \\ & \text{is the same form as} \qquad \qquad x^ 2 + bx + c\end{align*}

This means that we need to find two numbers m\begin{align*}m\end{align*} and n\begin{align*}n\end{align*} where

n+m=bandmn=c\begin{align*} n + m = b \qquad \qquad \text{and} \qquad \qquad mn = c\end{align*}

The factors of x2+bx+c\begin{align*}x^2 + bx + c\end{align*} are always two binomials

(x+m)(x+n)\begin{align*}(x + m)(x + n)\end{align*}

such that n+m=b\begin{align*}n + m = b\end{align*} and mn=c\begin{align*}mn = c\end{align*}.

#### Factoring

1. Factor x2+5x+6\begin{align*}x^2 + 5x + 6\end{align*}.

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

(x)(x)\begin{align*}(x\;\;\;\;)(x\;\;\;\;)\end{align*}

We want two numbers m\begin{align*}m\end{align*} and n\begin{align*}n\end{align*} that multiply to 6 and add up to 5. A good strategy is to list the possible ways we can multiply two numbers to get 6 and then see which of these numbers add up to 5:

6=166=23andand1+6=72+3=5This is the correct choice.\begin{align*}& 6 = 1 \cdot 6 && \text{and} && 1 + 6 = 7\\ & 6 = 2 \cdot 3 && \text{and} && 2 + 3 = 5 \qquad This \ is \ the \ correct \ choice.\end{align*}

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

We can check to see if this is correct by multiplying (x+2)(x+3)\begin{align*}(x + 2)(x + 3)\end{align*}:

x+2x+3 3x+6x2+2xx2+5x+6\begin{align*}& \quad \quad x + 2\\ & \underline{\;\;\;\;\;\;\;\;\;\;x + 3}\\ & \quad \quad \ 3x + 6\\ & \underline{x^2 + 2x\;\;\;\;\;\;}\\ & x^2 + 5x + 6\end{align*}

The answer checks out.

2. Factor x2+7x+12\begin{align*}x^2 + 7x + 12\end{align*}.

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

The number 12 can be written as the product of the following numbers:

12=11212=2612=34andandand1+12=132+6=83+4=7This is the correct choice.\begin{align*}& 12 = 1 \cdot 12 && \text{and} && 1 + 12 = 13\\ & 12 = 2 \cdot 6 && \text{and} && 2 + 6 = 8\\ & 12 = 3 \cdot 4 && \text{and} && 3 + 4 = 7 \qquad This \ is \ the \ correct \ choice.\end{align*}

The answer is (x+3)(x+4)\begin{align*}(x + 3)(x + 4)\end{align*}.

3. Factor x2+8x+12\begin{align*}x^2 + 8x + 12\end{align*}.

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

The number 12 can be written as the product of the following numbers:

12=11212=2612=34andandand1+12=132+6=8This is the correct choice.3+4=7\begin{align*}& 12 = 1 \cdot 12 && \text{and} && 1 + 12 = 13\\ & 12 = 2 \cdot 6 && \text{and} && 2 + 6 = 8 \qquad This \ is \ the \ correct \ choice.\\ & 12 = 3 \cdot 4 && \text{and} && 3 + 4 = 7\end{align*}

The answer is \begin{align*}(x + 2)(x + 6)\end{align*}.

### Example

#### Example 1

Factor \begin{align*}x^2 + 12x + 36\end{align*}.

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

The number 36 can be written as the product of the following numbers:

\begin{align*}& 36 = 1 \cdot 36 && \text{and} && 1 + 36 = 37\\ & 36 = 2 \cdot 18 && \text{and} && 2 + 18 = 20\\ & 36 = 3 \cdot 12 && \text{and} && 3 + 12 = 15\\ & 36 = 4 \cdot 9 && \text{and} && 4 + 9 = 13\\ & 36 = 6 \cdot 6 && \text{and} && 6 + 6 = 12 \qquad This \ is \ the \ correct \ choice.\end{align*}

The answer is \begin{align*}(x + 6)(x + 6)\end{align*}.

### Review

Factor the following quadratic polynomials.

1. \begin{align*}x^2 + 10x + 9\end{align*}
2. \begin{align*}x^2 + 15x + 50\end{align*}
3. \begin{align*}x^2 + 10x + 21\end{align*}
4. \begin{align*}x^2 + 16x + 48\end{align*}
5. \begin{align*}x^2+14x+45\end{align*}
6. \begin{align*}x^2+15x+50\end{align*}
7. \begin{align*}x^2+22x+40\end{align*}
8. \begin{align*}x^2+15x+56\end{align*}
9. \begin{align*}x^2+2x+1\end{align*}
10. \begin{align*}x^2+10x+24\end{align*}
11. \begin{align*}x^2+17x+72\end{align*}
12. \begin{align*}x^2+25x+150\end{align*}

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

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### Vocabulary Language: English

Greatest Common Factor

The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.

linear factors

Linear factors are expressions of the form $ax+b$ where $a$ and $b$ are real numbers.

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.

The quadratic formula states that for any quadratic equation in the form $ax^2+bx+c=0$, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

A quadratic polynomial is a polynomial of the 2nd degree, in other words, a polynomial with an $x^2$ term.

Trinomial

A trinomial is a mathematical expression with three terms.