About Factoring by Substitution:

We previously mastered factoring a polynomial of the form: ax2 + bx + c. In some cases, we will encounter a polynomial that is more complex but can be re-written through substitution. Once we perform the substitution, we factor as we normally do, then substitute one last time to obtain our final form.


Test Objectives
  • Demonstrate the ability to factor out the GCF or -(GCF) from a group of terms
  • Demonstrate the ability to re-write a polynomial using substitution
  • Demonstrate the ability to factor a polynomial using substitution
Factoring by Substitution Practice Test:

#1:

Instructions: Factor each.

$$a)\hspace{.2em}x^4 - 4$$

$$b)\hspace{.2em}x^4 + 3x^2$$


#2:

Instructions: Factor each.

$$a)\hspace{.2em}3x^7 + 31x^5 + 56x^3$$

$$b)\hspace{.2em}72x^8 + 42x^4 - 294$$


#3:

Instructions: Factor each.

$$a)\hspace{.2em}16x^8 - 28x^4 - 98$$

$$b)\hspace{.2em}16x^8 - 108x^4 + 162$$


#4:

Instructions: Factor each.

$$a)\hspace{.2em}18x^6 - 24x^3 + 8$$

$$b)\hspace{.2em}48x^8 - 4x^4 - 140$$


#5:

Instructions: Factor each.

$$a)\hspace{.2em}10(2x - 1)^2 - 19(2x - 1) - 15$$

$$b)\hspace{.2em}(3x + 5)^2 - 18(3x + 5) + 81$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}(x^2 - 2)(x^2 + 2)$$

$$b)\hspace{.2em}x^2(x^2 + 3)$$


#2:

Solutions:

$$a)\hspace{.2em}x^3(3x^2 + 7)(x^2 + 8)$$

$$b)\hspace{.2em}6(4x^4 - 7)(3x^4 + 7)$$


#3:

Solutions:

$$a)\hspace{.2em}2(2x^4 - 7)(4x^4 + 7)$$

$$b)\hspace{.2em}2(2x^2 - 3)(2x^2 + 3)(2x^4 - 9)$$


#4:

Solutions:

$$a)\hspace{.2em}2(3x^3 - 2)^2$$

$$b)\hspace{.2em}4(3x^4 + 5)(4x^4 - 7)$$


#5:

Solutions:

$$a)\hspace{.2em}2(5x - 1)(4x - 7)$$

$$b)\hspace{.2em}(3x - 4)^2$$