Sum And Difference of Cubic Polynomials
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Factoring two terms, both of them cubes does not come up very often in calculus. But you will see it. We include this discussion to give you a place to come back to when you need to remind yourself how to factor these types of equations.
\( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) |
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\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \) |
For a discussion and some good examples of the sum and difference of cubes, go to this Purple Math page .
Okay, try these practice problems.
Unless otherwise instructed, factor these cubic polynomials, giving your answers in exact form.
\( x^3 - 27 \)
Problem Statement
Factor \( x^3 - 27 \)
Solution
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\( 8x^3 - 64 \)
Problem Statement
Factor \( 8x^3 - 64 \)
Solution
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\( 8-t^3 \)
Problem Statement
Factor \( 8-t^3 \)
Solution
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\( 125A^3 + 27B^3 \)
Problem Statement
Factor \( 125A^3 + 27B^3 \)
Solution
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\( 64x^3 + 125 \)
Problem Statement
Factor the polynomial \( 64x^3 + 125 \)
Solution
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\( 27y^3 - 8 \)
Problem Statement
Factor \( 27y^3 - 8 \)
Solution
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\( 8x^3 + 27 \)
Problem Statement
Factor \( 8x^3 + 27 \)
Solution
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\( 27x^3 + 64y^3 \)
Problem Statement
Factor \( 27x^3 + 64y^3 \)
Solution
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\( 64y^3 - 125 \)
Problem Statement
Factor \( 64y^3 - 125 \)
Solution
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\( 8y^3-27 \)
Problem Statement
Factor the polynomial \( 8y^3-27 \)
Solution
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\( x^6 - 64y^9 \)
Problem Statement |
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Factor \( x^6 - 64y^9 \)
Final Answer |
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\( (x^2-4y^3)(x^4+4x^2y^3+16y^6) \)
Problem Statement
Factor \( x^6 - 64y^9 \)
Solution
Final Answer
\( (x^2-4y^3)(x^4+4x^2y^3+16y^6) \)
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\( (2x+3y)^3 - 343 \)
Problem Statement
Factor the polynomial \( (2x+3y)^3 - 343 \)
Solution
Not all instructors require students to multiply out the larger term. Check with your instructor to see what they expect.
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