2.1.1: Methods for Solving Quadratic Functions
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 14122
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}−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.
2x^{4}−x^{2}−15
2x^{4}−6x^{2}+5x^{2}−15
2x^{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}−16
Treat 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 realnumber solutions of 6x^{5}−51x^{3}−27x=0.
First, pull out the GCF among the three terms.
6x^{5}−51x^{3}−27x=0
3x(2x^{4}−17x^{2}−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.
6x^{5}−51x^{3}−27x=0
3x(2x^{4}−17x^{2}−9)=0
3x(2x^{4}−18x^{2}+x^{2}−9)=0
3x[2x^{2}(x^{2}−9)+1(x^{2}−9)]=0
3x(x^{2}−9)(2x^{2}+1)=0
Factor x^{2}−9 further and solve for x where possible. 2x^{2}+1 is not factorable.
3x(x^{2}−9)(2x^{2}+1)=0
3x(x−3)(x+3)(2x^{2}+1)=0
x=−3,0,3
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 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.
Example 2
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)
Example 3
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.
Example 4
Find all the realnumber 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 realnumber 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 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. 