2.1.1: Methods for Solving Quadratic Functions

Factoring Polynomials in Quadratic Form

The volume of a rectangular prism is 10x³−25x²−15x. What are the lengths of the prism's sides?

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Factoring Polynomials in Quadratic Form

The last type of factorable polynomial are those that are in quadratic form. Quadratic form is when a polynomial looks like a trinomial or binomial and can be factored like a quadratic. One example is when a polynomial is in the form ax⁴+bx²+c. Another possibility is something similar to the difference of squares, a⁴−b⁴. This can be factored to (a²−b²)(a²+b²) or (a−b)(a+b)(a²+b²). Always keep in mind that the greatest common factors should be factored out first.

Let's factor the following polynomials.

1. 2x⁴−x²−15

   This particular polynomial is factorable. First, ac=−30. The factors of -30 that add up to -1 are -6 and 5. Expand the middle term and then use factoring by grouping.

   2x⁴−x²−15

   2x⁴−6x²+5x²−15

   2x²(x²−3)+5(x²−3)

   (x²−3)(2x²+5)

   Both of the factors are not factorable, so we are done.

2. 81x⁴−16

   Treat this polynomial equation like a difference of squares.

   81x⁴−16

   (9x²−4)(9x²+4)

   Now, we can factor 9x²−4 using the difference of squares a second time.

   (3x−2)(3x+2)(9x²+4)

   9x²+4 cannot be factored because it is a sum of squares. This will have imaginary solutions.

   Now, let's find all the real-number solutions of 6x⁵−51x³−27x=0.

   First, pull out the GCF among the three terms.

   6x⁵−51x³−27x=0

   3x(2x⁴−17x²−9)=0

   Factor what is inside the parenthesis like a quadratic equation. ac=−18 and the factors of -18 that add up to -17 are -18 and 1. Expand the middle term and then use factoring by grouping.

   6x⁵−51x³−27x=0

   3x(2x⁴−17x²−9)=0

   3x(2x⁴−18x²+x²−9)=0

   3x[2x²(x²−9)+1(x²−9)]=0

   3x(x²−9)(2x²+1)=0

   Factor x²−9 further and solve for x where possible. 2x²+1 is not factorable.

   3x(x²−9)(2x²+1)=0

   3x(x−3)(x+3)(2x²+1)=0

   x=−3,0,3

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Examples

Example 1

Earlier, you were asked to find the lengths of the prism's sides.

Solution

To find the lengths of the prism's sides, we need to factor 10x³−25x²−15x.

First, pull out the GCF among the three terms.

10x³−25x²−15x

5x(2x²−5x−3)

Factor what is inside the parenthesis like a quadratic equation. ac=−6 and the factors of -6 that add up to -5 are -6 and 1.

5x(2x²−5x−3)=5x(2x+1)(x−3)

Therefore, the lengths of the rectangular prism's sides are 5x, 2x+1, and x−3.

Example 2

Factor: 3x⁴+14x²+8.

Solution

ac=24 and the factors of 24 that add up to 14 are 12 and 2.

3x⁴+14x²+8

3x⁴+12x²+2x²+8

3x²(x²+4)+2(x⁴+4)

(x²+4)(3x²+2)

Example 3

Factor: 36x⁴−25.

Solution

Factor this polynomial like a difference of squares.

36x⁴−25

(6x²−5)(6x²+5)

6 and 5 are not square numbers, so this cannot be factored further.

Example 4

Find all the real-number solutions of 8x⁵+26x³−24x=0.

Solution

Pull out a 2x from each term.

8x⁵+26x³−24x=0

2x(4x⁴+13x−12)=0

2x(4x⁴+16x²−3x²−12)=0

2x[4x²(x²+4)−3(x²+4)]=0

2x(x²+4)(4x²−3)=0

Set each factor equal to zero.

4x²−3=0

2x=0

x²+4=0

and x²= \(\ 3 \over 4\)

x=0

x²=−4

x=± \(\ \frac{\sqrt{3}}{2}\)

Notice the second factor will give imaginary solutions.

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Review

Factor the following quadratics completely.

1. x⁴−6x²+8
2. x⁴−4x²−45
3. x⁴−18x²+45
4. 4x⁴−11x²−3
5. 6x⁴+19x²+8
6. x⁴−81
7. 16x⁴−1
8. 6x⁵+26x³−20x
9. 4x⁶−36x²
10. 625−81x⁴

Find all the real-number solutions to the polynomials below.

11. 2x⁴−5x²−12=0
12. x⁴−16=0
13. 16x⁴−49=0
14. 12x⁶+69x⁴+45x²=0
15. 3x⁴+17x²−6=0

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Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.8.

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Vocabulary

Term	Definition
Factor to Solve	"Factor to Solve" is a common method for solving quadratic equations accomplished by factoring a trinomial into two binomials and identifying the values of x that make each binomial equal to zero.
factored form	The factored form of a quadratic function f(x) is f(x)=a(x−r1)(x−r2), where r1 and r2 are the roots of the function.
Factoring	Factoring is the process of dividing a number or expression into a product of smaller numbers or expressions.
Quadratic form	A polynomial in quadratic form looks like a trinomial or binomial and can be factored like a quadratic expression.
quadratic function	A quadratic function is a function that can be written in the form f(x)=ax^2+bx+c, where a, b, and c are real constants and a≠0.
Roots	The roots of a function are the values of x that make y equal to zero.
standard form	The standard form of a quadratic function is f(x)=ax^2+bx+c.
Vertex form	The vertex form of a quadratic function is y=a(x−h)^2+k, where (h,k) is the vertex of the parabola.
Zeroes of a Polynomial	The zeroes of a polynomial f(x) are the values of x that cause f(x) to be equal to zero.