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How Do You Factor a Polynomial Using Difference of Cubes?

Factor 27x6y3-8 completely

Summary

  1. The 'x' and 'y' are variables
  2. We want to use the difference of cubes formula, so try get everything into a cubed form
  3. The difference of cubes formula is: a3-b3=(a-b)(a2+ab+b2)
  4. The 'a' and 'b' refer to the variables in the sum of cubes formula
  5. Simply plugging in our values for 'a' and 'b' and simplifying will put our polynomial in factored form

Notes

    1. '27x6y3-8' is our polynomial
    2. The 'x' and 'y' are variables, and the '27' and '8' are constants
    1. '27x6y3-8' is our polynomial
    2. Our two terms are '27x6y3' and '8'
    1. '27x6y3-8' is our polynomial
    2. Our two terms are '27x6y3' and '8'
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    1. '27x6y3-8' is our polynomial
    2. Our two terms are '27x6y3' and '8'
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    1. '27x6y3-8' is our polynomial
    2. Our two terms are '27x6y3' and '8'
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    4. The 'x' and 'y' are variables, and the '27' and '8' are constants
    5. We're trying to rewrite our polynomial so it looks like 'a3-b3' in the difference of cubes formula
    1. '27x6y3-8' is our polynomial
    2. Our two terms are '27x6y3' and '8'
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    4. We're trying to rewrite our polynomial so it looks like 'a3-b3' in the difference of cubes formula
    5. 33=27
    1. '27x6y3-8' is our polynomial
    2. Our two terms are '27x6y3' and '8'
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    4. We're trying to rewrite our polynomial so it looks like 'a3-b3' in the difference of cubes formula
    5. 33=27
    6. (x2)3=x6
    1. '27x6y3-8' is our polynomial
    2. Our two terms are '27x6y3' and '8'
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    4. We're trying to rewrite our polynomial so it looks like 'a3-b3' in the difference of cubes formula
    5. 33=27
    6. (x2)3=x6
    7. y3=y3
    1. '27x6y3-8' is our polynomial
    2. Our two terms are '27x6y3' and '8'
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    4. We're trying to rewrite our polynomial so it looks like 'a3-b3' in the difference of cubes formula
    5. 33=27
    6. (x2)3=x6
    7. y3=y3
    8. 23=8
    1. Our first term is now: '33(x3)3y3'
    2. Our second term is now: '(2)3'
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    4. We're trying to rewrite our polynomial so it looks like 'a3-b3' in the difference of cubes formula
    1. Up to this point, our first term is: '33(x3)3y3'
    2. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    3. We're trying to rewrite our polynomial so it looks like 'a3-b3' in the difference of cubes formula
    1. Up to this point, our first term is: '33(x3)3y3'
    2. '33(x3)3y3' can be rewritten as '(3x2y)3'
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    4. We're trying to rewrite our polynomial so it looks like 'a3-b3' in the difference of cubes formula
    1. Up to this point, our second term is: '(2)y3'
    2. So we can keep it as is
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    4. We're trying to rewrite our polynomial so it looks like 'a3-b3' in the difference of cubes formula
    1. Our polynomial has been rewritten as: '(3x2y)3-(2)3'
    1. Our polynomial has been rewritten as: '(3x2y)3-(2)3'
    2. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    1. Our polynomial has been rewritten as: '(3x2y)3-(2)3'
    2. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    1. Our polynomial has been rewritten as: '(3x2y)3-(2)3'
    2. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    1. Our polynomial has been rewritten as: '(3x2y)3-(2)3'
    2. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    3. Our 'a' term is '3x2y'
    1. Our polynomial has been rewritten as: '(3x2y)3-(2)3'
    2. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    3. Our 'a' term is '3x2y'
    1. Our polynomial has been rewritten as: '(3x2y)3-(2)3'
    2. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    3. Our 'a' term is '3x2y'
    4. Our 'b' term is '2'
    1. Our polynomial has been rewritten as: '(3x2y)3-(2)3'
    2. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    3. Our 'a' term is '3x2y'
    4. Our 'b' term is '2'
    1. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    1. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    1. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    2. Plugging in our values for 'a' and 'b' into the difference of cubes formula and simplifying will factor our polynomial for us!
    1. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    2. Plugging in our values for 'a' and 'b' into the difference of cubes formula and simplifying will factor our polynomial for us!
    1. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    2. Plugging in our values for 'a' and 'b' into the difference of cubes formula and simplifying will factor our polynomial for us!
    3. '(a-b)=(3x2y-2)'
    1. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    2. Plugging in our values for 'a' and 'b' into the difference of cubes formula and simplifying will factor our polynomial for us!
    3. '(a-b)=(3x2y-2)'
    4. 'a2=(3x2y)2'
    1. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    2. Plugging in our values for 'a' and 'b' into the difference of cubes formula and simplifying will factor our polynomial for us!
    3. '(a-b)=(3x2y-2)'
    4. 'a2=(3x2y)2'
    5. 'ab=(3x2y)(2)'
    1. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    2. Plugging in our values for 'a' and 'b' into the difference of cubes formula and simplifying will factor our polynomial for us!
    3. '(a-b)=(3x2y-2)'
    4. 'a2=(3x2y)2'
    5. 'ab=(3x2y)(2)'
    6. 'b2=(2)2'
    1. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'
    2. Plugging in our values for 'a' and 'b' into the difference of cubes formula and simplifying will factor our polynomial for us!
    3. '(a-b)=(3x2y-2)'
    4. 'a2=(3x2y)2'
    5. 'ab=(3x2y)(2)'
    6. 'b2=(2)2'
    1. So we need to simplify: '(3x2-2)([3x2y)2]+3x2y•2+22)'
    1. So we need to simplify: '(3x2-2)([3x2y)2]+3x2y•2+22)'
    2. [3x2y)2]=9x4y2
    3. 3x2y•2=6x2y
    4. 22=4
    1. We factored '27x6y3-8' into '(3x2y-2)(9x4y2+6x2y+4)'
    1. We factored '27x6y3-8' into '(3x2y-2)(9x4y2+6x2y+4)'
    2. Our two terms were '27x6y3' and '8'
    1. We factored '27x6y3-8' into '(3x2y-2)(9x4y2+6x2y+4)'
    2. Our two terms were '27x6y3' and '8'
    3. The difference of cubes formula is 'a3-b3=(a-b)(a2+ab+b2)'