9.5: Factoring Quadratic Expressions
In this lesson, we will learn how to factor quadratic polynomials for different values of , and . In the last lesson, we factored common monomials, so you already know how to factor quadratic polynomials where .
Factoring Quadratic Expressions in Standard From
Quadratic polynomials are polynomials of degree 2. The standard form of a quadratic polynomial is , where , and are real numbers.
Example 1: Factor .
Solution: We are looking for an answer that is a product of 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. The number six can be written as the product of.
So the answer is .
We can check to see if this is correct by multiplying .
is multiplied by and .
2 is multiplied by and .
Combine the like terms: .
Example 2: Factor .
Solution: We are looking for an answer that is a product of the two parentheses .
The number 8 can be written as the product of the following numbers.
and Notice that these are two different choices.
And
and This is the correct choice.
The answer is .
Example 3: Factor .
Solution: We are looking for an answer that is a product of two 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 Notice that these are two different choices.
And also,
and Notice that these are two different choices.
and This is the correct choice.
and
The answer is .
Example 4: Factor .
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.
We are looking for an answer that is a product of two parentheses .
Now our job is to factor .
The number –6 can be written as the product of the following numbers.
The answer is .
To Summarize:
A quadratic of the form factors as a product of two parenthesis .
- 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 .
Practice Set
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.
Mixed Review
- Evaluate when .
- The Nebraska Department of Roads collected the following data regarding mobile phone distractions in traffic crashes by teen drivers.
- Plot the data as a scatter plot.
- Fit a line to this data.
- Predict the number of teenage traffic accidents attributable to cell phones in the year 2012.
Year () | Total () |
---|---|
2002 | 41 |
2003 | 43 |
2004 | 47 |
2005 | 38 |
2006 | 36 |
2007 | 40 |
2008 | 42 |
2009 | 42 |
- Simplify .
- Graph the following on a number line: .
- What is the multiplicative inverse of ?
Quick Quiz
- Name the following polynomial. State its degree and leading coefficient .
- Simplify .
- A rectangular solid has dimensions by by . Find its volume.
- Simplify .
- Find the solutions to .
- Multiply .