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6.8: Factoring Polynomials in Quadratic Form

Difficulty Level: At Grade Created by: CK-12
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The volume of a rectangular prism is 10x325x215x. What are the lengths of the prism's sides?

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 ax4+bx2+c. Another possibility is something similar to the difference of squares, a4b4. This can be factored to (a2b2)(a2+b2) or (ab)(a+b)(a2+b2). Always keep in mind that the greatest common factors should be factored out first.

Example A

Factor 2x4x215.

Solution: This particular polynomial is factorable. Let’s use the method we learned in the Factoring when a1 concept. 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.

2x4x2152x46x2+5x2152x2(x23)+5(x23)(x23)(2x2+5)

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

Example B

Factor 81x416.

Solution: Treat this polynomial equation like a difference of squares.

81x416(9x24)(9x2+4)

Now, we can factor 9x24 using the difference of squares a second time.

(3x2)(3x+2)(9x2+4)

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

Example C

Find all the real-number solutions of 6x551x327x=0.

Solution: First, pull out the GCF among the three terms.

6x551x327x3x(2x417x29)=0=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.

6x551x327x3x(2x417x29)3x(2x418x2+x29)3x[2x2(x29)+1(x29)]3x(x29)(2x2+1)=0=0=0=0=0

Factor x29 further and solve for x where possible. 2x2+1 is not factorable.

3x(x29)(2x2+1)3x(x3)(x+3)(2x2+1)x=0=0=3,0,3

Intro Problem Revisit To find the lengths of the prism's sides, we need to factor 10x325x215x.

First, pull out the GCF among the three terms.

10x325x215x5x(2x25x3)

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(2x25x3)=5x(2x+1)(x3)

Therefore, the lengths of the rectangular prism's sides are 5x, 2x+1, and x3.

Guided Practice

Factor the following polynomials.

1. 3x4+14x2+8

2. 36x425

3. Find all the real-number solutions of 8x5+26x324x=0.

Answers

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

3x4+14x2+83x4+12x2+2x2+83x2(x2+4)+2(x4+4)(x2+4)(3x2+2)

2. Factor this polynomial like a difference of squares.

36x425(6x25)(6x2+5)

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

3. Pull out a 2x from each term.

8x5+26x324x2x(4x4+13x12)2x(4x4+16x23x212)2x[4x2(x2+4)3(x2+4)]2x(x2+4)(4x23)=0=0=0=0=0

Set each factor equal to zero.

  4x23=02x=0x2+4=0      andx2=34 x=0x2=4 x=±32

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.

  1. x46x2+8
  2. x44x245
  3. x49x2+45
  4. 4x411x23
  5. 6x4+19x2+8
  6. x481
  7. 16x41
  8. 6x5+26x320x
  9. 4x636x2
  10. 62581x4

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

  1. 2x45x212=0
  2. x416=0
  3. \begin{align*}16x^4-49 = 0\end{align*}
  4. \begin{align*}12x^6+69x^4+45x^2 = 0\end{align*}
  5. \begin{align*}3x^4+17x^2 -6=0\end{align*}

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Vocabulary

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\ne 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.

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