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
#1:
Instructions: Factor each.
$$a)\hspace{.2em}x^4 - 4$$
$$b)\hspace{.2em}6x^{\frac{2}{3}}- 7x^{\frac{1}{3}}- 3$$
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#2:
Instructions: Factor each.
$$a)\hspace{.2em}3x^7 + 31x^5 + 56x^3$$
$$b)\hspace{.2em}72x^8 + 42x^4 - 294$$
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#3:
Instructions: Factor each.
$$a)\hspace{.2em}16x^8 - 28x^4 - 98$$
$$b)\hspace{.2em}16x^8 - 108x^4 + 162$$
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#4:
Instructions: Factor each.
$$a)\hspace{.2em}18x^6 - 24x^3 + 8$$
$$b)\hspace{.2em}48x^8 - 4x^4 - 140$$
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#5:
Instructions: Factor each.
$$a)\hspace{.2em}10(2x - 1)^2 - 19(2x - 1) - 15$$
$$b)\hspace{.2em}(3x + 5)^2 - 18(3x + 5) + 81$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}(x^2 - 2)(x^2 + 2)$$
$$b)\hspace{.2em}(3x^{\frac{1}{3}}+ 1)(2x^{\frac{1}{3}}- 3)$$
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#2:
Solutions:
$$a)\hspace{.2em}x^3(3x^2 + 7)(x^2 + 8)$$
$$b)\hspace{.2em}6(4x^4 - 7)(3x^4 + 7)$$
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#3:
Solutions:
$$a)\hspace{.2em}2(2x^4 - 7)(4x^4 + 7)$$
$$b)\hspace{.2em}2(2x^2 - 3)(2x^2 + 3)(2x^4 - 9)$$
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#4:
Solutions:
$$a)\hspace{.2em}2(3x^3 - 2)^2$$
$$b)\hspace{.2em}4(3x^4 + 5)(4x^4 - 7)$$
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#5:
Solutions:
$$a)\hspace{.2em}2(5x - 1)(4x - 7)$$
$$b)\hspace{.2em}(3x - 4)^2$$
