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Notes

Problems

Factor

\(\textbf{1)}\) \( x^3-1000 \)

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The answer is \( (x-10)(x^2+10x+100) \)

\(\textbf{2)}\) \( x^3-1 \)

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The answer is \( (x-1)(x^2+x+1) \)

\(\textbf{3)}\) \( y^6+1 \)

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The answer is \( (y^2+1)(y^4-y^2+1) \)

\(\textbf{4)}\) \( 16x^4-54x \)

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The answer is \( 2x(2x-3)(4x^2+6x+9) \)

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\(\text{Step 1: Factor out a GCF}\)

\(\,\,\,\,\,\,2x\left(8x^3-27\right)\)

\(\text{Step 2: Use the difference of two cubes equation}\)

\(\,\,\,\,\,\,\left(a^3-b^3\right)=\left(a-b\right)\left(a^2+ab+b^2\right)\)

\(\,\,\,\,\,\,a=\sqrt[3]{8x^3}=2x, \,\,\, b=\sqrt[3]{27}=3\)

\(\,\,\,\,\,\,\left(\left(2x\right)^3-\left(3\right)^3\right)=\left(2x-3\right)\left(\left(2x\right)^2+\left(2x\right)\left(3\right)+\left(3\right)^2\right)\)

\(\,\,\,\,\,\,\left(\left(2x\right)^3-\left(3\right)^3\right)=\left(2x-3\right)(4x^2+6x+9)\)

\(\text{Step 3: Bring down the GCF}\)

\(\,\,\,\,\,\,2x\left(2x-3\right)(4x^2+6x+9)\)

\(\textbf{5)}\) \( w^3-27y^9 \)

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The answer is \( (w-3y^3)(w^2+3wy^3+9y^6) \)

\(\textbf{6)}\) \( x^3+64 \)

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The answer is \( (x+4)(x^2-4x+16) \)

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\(\text{Use the sum of two cubes equation}\)

\(\,\,\,\,\,\,\left(a^3+b^3\right)=\left(a+b\right)\left(a^2-ab+b^2\right)\)

\(\,\,\,\,\,\,a=\sqrt[3]{x^3}=x, \,\,\, b=\sqrt[3]{64}=4\)

\(\,\,\,\,\,\,\left(\left(x\right)^3+\left(4\right)^3\right)=\left(x+4\right)\left(\left(x\right)^2-\left(4\right)\left(x\right)+\left(4\right)^2\right)\)

\(\,\,\,\,\,\,\left(\left(x\right)^3+\left(4\right)^3\right)=\left(x+4\right)\left(x^2-4x+16\right)\)

The answer is \( (x+4)(x^2-4x+16) \)

\(\textbf{7)}\) \( w^{12}+64y^3 \)

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The answer is \( (w^4+4y)(w^8-4w^4 y+16y^2) \)

\(\textbf{8)}\) \( x^3-8 \)

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The answer is \( (x-2)(x^2+2x+4) \)

Challenge Problems

\(\textbf{9)}\) Factor \( x^6-64 \) completely

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The answer is \( \left(x+2\right)\left(x-2\right)\left(x^2+2x+4\right)\left(x^2-2x+4\right) \)

\(\textbf{10)}\) Factor \( x^6-1 \) completely

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The answer is \( \left(x+1\right)\left(x-1\right)\left(x^2+x+1\right)\left(x^2-x+1\right) \)

See Related Pages\(\)

\(\bullet\text{ Factoring Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)

\(\bullet\text{ Factoring out a GCF}\)
\(\,\,\,\,\,\,\,\,3xyz^2+x^2y^2z+9x^3y=xy(3z^2+xyz+9x^2)…\)

\(\bullet\text{ Perfect Square Trinomials}\)
\(\,\,\,\,\,\,\,\,x^2-6x+9=(x-3)^2…\)

\(\bullet\text{ Factoring Trinomials with a}=1\)
\(\,\,\,\,\,\,\,\,x^2+7x+12=(x+3)(x+4)…\)

\(\bullet\text{ Factoring Trinomials with a} \ne 1\)
\(\,\,\,\,\,\,\,\,3x^2+11x+6=(3x+2)(x+3)…\)

\(\bullet\text{ Factoring with u-substitution}\)
\(\,\,\,\,\,\,\,\,x^4+5x^2+6=u^2+5u+6…\)

\(\bullet\text{ Difference of Two Squares}\)
\(\,\,\,\,\,\,\,\,x^2-16=(x+4)(x-4)…\)

\(\bullet\text{ Factor by Grouping}\)
\(\,\,\,\,\,\,\,\,8x^3-4x^2-6x+3=(4x^2-3)(2x-1)…\)

\(\bullet\text{ Solving Quadratic Equations by Factoring}\)
\(\,\,\,\,\,\,\,\,x^2+10x−24=0…\)

In Summary

Factoring is the process of expressing a number or algebraic expression as the product of other simpler numbers or expressions. The general formula for factoring the sum or difference of two cubes is as follows:

\(a^3 + b^3 = (a + b)(a^2 – ab + b^2)\)
\(a^3 – b^3 = (a – b)(a^2 + ab + b^2)\)

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