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## Factor quadratics with positive coefficients

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Suppose your height above sea level in feet 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. If you wanted to factor this expression, could you do it? What would be the steps that you would follow?

Previously, we factored common monomials, so you already know how to factor quadratic polynomials where \begin{align*}c=0\end{align*}. Now, we will learn how to factor quadratic polynomials for different values of \begin{align*}a,\ b\end{align*}, and \begin{align*}c\end{align*}.

Quadratic polynomials are polynomials of degree 2. The standard form of a quadratic polynomial is \begin{align*}ax^2+bx+c\end{align*}, where \begin{align*}a,\ b\end{align*}, and \begin{align*}c\end{align*} are real numbers.

A quadratic of the form \begin{align*}x^2+bx+c\end{align*} factors to the product of two binomials: \begin{align*}(x+m)(x+n)\end{align*} where \begin{align*}m+n = b\end{align*} and \begin{align*}mn = c\end{align*}.

There are some important rules to know when factoring quadratics:

• If \begin{align*}b\end{align*} and \begin{align*}c\end{align*} are positive then both \begin{align*}m\end{align*} and \begin{align*}n\end{align*} are positive.
• Example: \begin{align*}x^2+8x+12\end{align*} factors as \begin{align*}(x+6)(x+2)\end{align*}.
• If \begin{align*}b\end{align*} is negative and \begin{align*}c\end{align*} 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*}.

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

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

To fill in the blanks, we want two numbers \begin{align*}m\end{align*} and \begin{align*}n\end{align*} that multiply to 6 and add to 5. A good strategy is to list the possible ways we can multiply two numbers to give us 6 and then see which of these pairs of numbers add to 5.

Since both \begin{align*}b\end{align*} and \begin{align*}c\end{align*} are positive, then both spaces will be filled by positive numbers.

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

\begin{align*}6&=1 \times 6 \qquad and \qquad 1+6=7\\ 6&=2 \times 3 \qquad and \qquad 2+3=5\end{align*}

So the answer is \begin{align*}(x+2)(x+3)\end{align*} because 2 and 3 multiply to 6 and sum to 5.

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

\begin{align*}x\end{align*} is multiplied by \begin{align*}x\end{align*} and \begin{align*}3 \text{ to get }x^2+3x\end{align*}.

2 is multiplied by \begin{align*}x\end{align*} and \begin{align*}3 \text{ to get }2x+6\end{align*}.

Combine the like terms: \begin{align*}x^2+5x+6\end{align*}.

1. \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*}.

1. \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

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

Mixed Review

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

#### Quick Quiz

1. 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*}.
2. Simplify \begin{align*}(a^2 b^2 c+11abc^5 )+(4abc^5-3a^2 b^2 c+9abc)\end{align*}.
3. 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.
4. Simplify \begin{align*}-3hjk^3 (h^2 j^4 k+6hk^2)\end{align*}.
5. Find the solutions to \begin{align*}(x-3)(x+4)(2x-1)=0\end{align*}.
6. Multiply \begin{align*}(a-9b)(a+9b)\end{align*}.

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

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Color Highlighted Text Notes

### Vocabulary Language: English Spanish

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

The standard form of a quadratic polynomial is $ax^2+bx+c$, where $a,\ b$, and$c$ are real numbers.

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 $ax^2+bx+c=0$, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

Trinomial

A trinomial is a mathematical expression with three terms.