Math 2 (L1-7)
A.SSE.2, A.SSE.3a, A.REI.4
Assessment Title: 4th Degree Quadratic Form
Unit 3: Quadratic Functions: Working with Equations
Learning Targets:
 Factor a power function that models a quadratic function given in standard form
 Solve a power function equation that models a quadratic equation
We have been factoring quadratic expressions. Now, let’s extend our factoring skills to factor higher degree polynomials
that model quadratic form.
Quadratic Expression
Factor x2 - 8x + 16
(x - 4)(x - 4)
4th Degree Polynomial in Quadratic Form
Factor x4 - 8x2 + 16
(x2)2 - 8x2 + 16
Factor into two binomials.
Notice the 4th degree expression models a similar for as the quadratic
above.
Recall that x4 = (x2)2, and rewrite in quadratic form.
Sometimes, we define x2 by another variable, like u, and let u = x2. u is
substituted in for x2 in this step. u2 – 8u + 16.
(x2 - 4)(x2 - 4)
Factor into two binomials.
When using u substitution method, the expression would factor into (u –
4)(u – 4)
At this point, substitute x2 back in for u: (x2 - 4)(x2 - 4)
(x – 2)(x + 2)(x – 2)(x + 2)
Factor completely.
Factor using quadratic form.
1.
x4 – 2x2 + 1
2.
x4 – 16
3.
x4 + 6x2 + 5
4.
x4 - 7x2 + 12
5.
2x4 + 7x2 + 5
6.
x4 – 81
Math 2 (L1-7)
A.SSE.2, A.SSE.3a, A.REI.4
Assessment Title: 4th Degree Quadratic Form
Unit 3: Quadratic Functions: Working with Equations
ANSWER KEY
We have been factoring quadratic expressions. Now, let’s extend our factoring skills to factor higher degree polynomials
that model quadratic form.
Quadratic Expression
Factor x2 - 8x + 16
(x - 4)(x - 4)
4th Degree Polynomial in Quadratic Form
Factor x4 - 8x2 + 16
(x2)2 - 8x2 + 16
Factor into two binomials.
Notice the 4th degree expression models a similar for as the quadratic
above.
Recall that x4 = (x2)2, and rewrite in quadratic form.
Sometimes, we define x2 by another variable, like u, and let u = x2. u is
substituted in for x2 in this step. u2 – 8u + 16.
(x2 - 4)(x2 - 4)
Factor into two binomials.
When using u substitution method, the expression would factor into (u –
4)(u – 4)
At this point, substitute x2 back in for u: (x2 - 4)(x2 - 4)
(x – 2)(x + 2)(x – 2)(x + 2)
Factor completely.
Factor using quadratic form.
1.
x4 – 2x2 + 1
2.
(x2 – 1)(x2 – 1) = (x + 1)(x – 1)(x + 1)(x – 1)
x4 – 16
(x2 – 4)(x2 + 4) = (x + 2)(x – 2)(x2 + 4)
= (x + 1)2(x – 1)2
3.
x4 + 6x2 + 5
4.
(x2 + 5)(x2 + 1)
5.
2x4 + 7x2 + 5
(2x2 + 5)(x2 + 1)
x4 - 7x2 + 12
(x2 – 6)(x2 – 1) = (x2 – 6)(x + 1)(x - 1)
6.
x4 – 81
(x2 + 9)(x2 – 9) = (x2 + 9)(x + 3)(x – 3)