6.8: Factoring Polynomials in Quadratic Form
The volume of a rectangular prism is
Guidance
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
Example A
Factor
Solution: This particular polynomial is factorable. Let’s use the method we learned in the Factoring when
Both of the factors are not factorable, so we are done.
Example B
Factor
Solution: Treat this polynomial equation like a difference of squares.
Now, we can factor
Example C
Find all the realnumber solutions of
Solution: First, pull out the GCF among the three terms.
Factor what is inside the parenthesis like a quadratic equation.
Factor
Intro Problem Revisit To find the lengths of the prism's sides, we need to factor
First, pull out the GCF among the three terms.
Factor what is inside the parenthesis like a quadratic equation.
Therefore, the lengths of the rectangular prism's sides are
Guided Practice
Factor the following polynomials.
1.
2.
3. Find all the realnumber solutions of
Answers
1.
2. Factor this polynomial like a difference of squares.
6 and 5 are not square numbers, so this cannot be factored further.
3. Pull out a
Set each factor equal to zero.
Notice the second factor will give imaginary solutions.
Vocabulary
 Quadratic form
 When a polynomial looks a trinomial or binomial and can be factored like a quadratic equation.
Practice
Factor the following quadratics completely.

x4−6x2+8 
x4−4x2−45 
x4−9x2+45 
4x4−11x2−3 
6x4+19x2+8 
x4−81 
16x4−1 
6x5+26x3−20x 
4x6−36x2 
625−81x4
Find all the realnumber solutions to the polynomials below.

2x4−5x2−12=0 
x4−16=0  \begin{align*}16x^449 = 0\end{align*}
 \begin{align*}12x^6+69x^4+45x^2 = 0\end{align*}
 \begin{align*}3x^4+17x^2 6=0\end{align*}
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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 that make each binomial equal to zero.factored form
The factored form of a quadratic function is , where and 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 , where , , and are real constants and .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 .Vertex form
The vertex form of a quadratic function is , where is the vertex of the parabola.Zeroes of a Polynomial
The zeroes of a polynomial are the values of that cause to be equal to zero.Image Attributions
Here you'll how to factor and solve polynomials that are in “quadratic form.”