# 2.1.1: Methods for Solving Quadratic Functions

- Page ID
- 14122

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## Factoring Polynomials in Quadratic Form

The volume of a rectangular prism is 10x^{3}−25x^{2}−15x. What are the lengths of the prism's sides?

## 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 ax^{4}+bx^{2}+c. Another possibility is something similar to the difference of squares, a^{4}−b^{4}. This can be factored to (a^{2}−b^{2})(a^{2}+b^{2}) or (a−b)(a+b)(a^{2}+b^{2}). Always keep in mind that the greatest common factors should be factored out first.

Let's factor the following polynomials.

- 2x
^{4}−x^{2}−15This 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.

2x

^{4}−x^{2}−152x

^{4}−6x^{2}+5x^{2}−152x

^{2}(x^{2}−3)+5(x^{2}−3)(x

^{2}−3)(2x^{2}+5)Both of the factors are not factorable, so we are done.

- 81x
^{4}−16Treat this polynomial equation like a difference of squares.

81x

^{4}−16(9x

^{2}−4)(9x^{2}+4)Now, we can factor 9x

^{2}−4 using the difference of squares a second time.(3x−2)(3x+2)(9x

^{2}+4)9x

^{2}+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 6x

^{5}−51x^{3}−27x=0.First, pull out the GCF among the three terms.

6x

^{5}−51x^{3}−27x=03x(2x

^{4}−17x^{2}−9)=0Factor 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.

6x

^{5}−51x^{3}−27x=03x(2x

^{4}−17x^{2}−9)=03x(2x

^{4}−18x^{2}+x^{2}−9)=03x[2x

^{2}(x^{2}−9)+1(x^{2}−9)]=03x(x

^{2}−9)(2x^{2}+1)=0Factor x

^{2}−9 further and solve for x where possible. 2x^{2}+1 is not factorable.3x(x

^{2}−9)(2x^{2}+1)=03x(x−3)(x+3)(2x

^{2}+1)=0x=−3,0,3

## Examples

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 10x^{3}−25x^{2}−15x.

First, pull out the GCF among the three terms.

10x^{3}−25x^{2}−15x

5x(2x^{2}−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(2x^{2}−5x−3)=5x(2x+1)(x−3)

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

Factor: 3x^{4}+14x^{2}+8.

**Solution**

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

3x^{4}+14x^{2}+8

3x^{4}+12x^{2}+2x^{2}+8

3x^{2}(x^{2}+4)+2(x^{4}+4)

(x^{2}+4)(3x^{2}+2)

Factor: 36x^{4}−25.

**Solution**

Factor this polynomial like a difference of squares.

36x^{4}−25

(6x^{2}−5)(6x^{2}+5)

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

Find all the real-number solutions of 8x^{5}+26x^{3}−24x=0.

**Solution**

Pull out a 2x from each term.

8x^{5}+26x^{3}−24x=0

2x(4x^{4}+13x−12)=0

2x(4x^{4}+16x^{2}−3x^{2}−12)=0

2x[4x^{2}(x^{2}+4)−3(x^{2}+4)]=0

2x(x^{2}+4)(4x^{2}−3)=0

Set each factor equal to zero.

4x^{2}−3=0

2x=0

x^{2}+4=0

and x^{2}= \(\ 3 \over 4\)

x=0

x^{2}=−4

x=± \(\ \frac{\sqrt{3}}{2}\)

Notice the second factor will give imaginary solutions.

## Review

Factor the following quadratics completely.

- x
^{4}−6x^{2}+8 - x
^{4}−4x^{2}−45 - x
^{4}−18x^{2}+45 - 4x
^{4}−11x^{2}−3 - 6x
^{4}+19x^{2}+8 - x
^{4}−81 - 16x
^{4}−1 - 6x
^{5}+26x^{3}−20x - 4x
^{6}−36x^{2} - 625−81x
^{4}

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

- 2x
^{4}−5x^{2}−12=0 - x
^{4}−16=0 - 16x
^{4}−49=0 - 12x
^{6}+69x^{4}+45x^{2}=0 - 3x
^{4}+17x^{2}−6=0

## 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−r_{1})(x−r_{2}), where r_{1} and r_{2} 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 that make x equal to zero.y |

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. |