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Factorization of Quadratic Expressions

Factor quadratics with positive coefficients

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Factorization of Quadratic Expressions

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

Factoring Quadratic Expressions

Previously, we factored common monomials, so you already know how to factor quadratic polynomials where . Now, we will learn how to factor quadratic polynomials for different values of , and .

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

A quadratic of the form factors to the product of two binomials:  where  and .

There are some important rules to know when factoring quadratics: 

  • If and are positive then both and  are positive.
    • Example: factors as .
  • If is negative and is positive then both and  are negative.
    • Example: factors as .
  • If is negative then either is positive and  is negative or vice-versa.
    • Example: factors as .
    • Example: factors as .
  • If , factor a common factor of –1 from each term in the trinomial and then factor as usual. The answer will have the form .
    • Example: factors as .

Let's factor the following polynomials:

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 and 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  and  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:

So the answer is  because 2 and 3 multiply to 6 and sum to 5.

We can check to see if this is correct by multiplying .

is multiplied by and .

2 is multiplied by and .

Combine the like terms: .

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

 Since  is negative and  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:

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

The answer is .

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

Since  is positive and  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:

 -3 and 5 are the factors that multiply to -15 and add to 2. The answer is .

Examples

Example 1

Earlier, you were told that your height above sea level when traveling over a hill can be represented by the expression , where  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.

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

Now our job is to factor . Since  is negative and  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:

3 and -21 are the factors that multiply to -63 and sum to -18.The answer is .

Example 2

Factor .

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

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

Now our job is to factor .

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

-3 and 2 are the factors that multiply to -6 and add to -1. The answer is .

Review

Factor the following quadratic polynomials.

Mixed Review

  1. Evaluate when .
  2. Simplify .
  3. Graph the following on a number line: .
  4. What is the multiplicative inverse of ?

Quick Quiz

  1. Name the following polynomial. State its degree and leading coefficient: .
  2. Simplify .
  3. A rectangular solid has dimensions by by . Find its volume.
  4. Simplify .
  5. Find the solutions to .
  6. Multiply .

Review (Answers)

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

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Vocabulary

TermDefinition
Quadratic Polynomials A quadratic polynomial is a polynomial of the 2nd degree, in other words, a polynomial with an x^2 term.
standard form of quadratic polynomials The standard form of a quadratic polynomial is ax^2+bx+c, where a,\ b, andc 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.
Quadratic Formula 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.

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