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How Do You Factor a Polynomial Using Sum of Cubes?

Factor 27x6y3+8 completely

Summary

  1. The 'x' and 'y' are variables
  2. We want to use the sum of cubes formula, so try get everything into a cubed form
  3. "Cubed" is another way of saying "to the 3rd power"
  4. The sum of cubes formula is: a3+b3=(a+b)(a2-ab+b2)
  5. The 'a' and 'b' refer to the variables in the sum of cubes formula
  6. 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. Our two terms are '27x6y3' and '8'
    1. '27x6y3+8' is our polynomial
    2. Our two terms are '27x6y3' and '8'
    1. '27x6y3+8' is our polynomial
    1. '27x6y3+8' is our polynomial
    2. "Cubed" is another way of saying "to the 3rd power"
    1. '27x6y3+8' is our polynomial
    2. 33=27
    3. "Cubed" is another way of saying "to the 3rd power"
    1. '27x6y3+8' is our polynomial
    2. 33=27
    3. (x2)3=x6
    1. '27x6y3+8' is our polynomial
    2. 33=27
    3. (x2)3=x6
    4. y3=y3
    5. "Cubed" is another way of saying "to the 3rd power"
    1. '27x6y3+8' is our polynomial
    2. 33=27
    3. (x2)3=x6
    4. y3=y3
    5. 23=8
    1. '27x6y3+8' is our polynomial
    2. 33=27
    3. (x2)3=x6
    4. y3=y3
    5. 23=8
    6. "Cubed" is another way of saying "to the 3rd power"
    1. '27x6y3+8' is our polynomial
    2. 33=27
    3. (x2)3=x6
    4. y3=y3
    5. 23=8
    1. '27x6y3+8' is our polynomial
    2. Our first term is now: '33(x3)3y3'
    3. "Cubed" is another way of saying "to the 3rd power"
    1. '27x6y3+8' is our polynomial
    2. Our first term is now: '33(x3)3y3'
    3. '33(x3)3y3' can be rewritten as '(3x2y)3'
    1. '27x6y3+8' is our polynomial
    2. Our first term is now: '33(x3)3y3'
    3. '33(x3)3y3' can be rewritten as '(3x2y)3'
    4. Our second term is already cubed: '(2)3'
    1. '27x6y3+8' was our original polynomial
    2. This can be rewritten as: '(3x2y)3+(2)3'
    3. This format resembles the formula for the sum of cubes!
    4. "Cubed" is another way of saying "to the 3rd power"
    1. Our polynomial has been rewritten as: '(3x2y)3+(2)3'
    2. It still has two terms, which we can label as 'a' and b'
    3. Labeling the terms as 'a' and 'b' will make it easier to compare our polynomial to the sum of cubes formula
    1. Our polynomial has been rewritten as: '(3x2y)3+(2)3'
    2. Labeling the terms as 'a' and 'b' will make it easier to compare our polynomial to the sum of cubes formula
    3. Our 'a' term is '3x2y'
    4. "Cubing" is another way of saying "taking something to the 3rd power"
    1. Our polynomial has been rewritten as: '(3x2y)3+(2)3'
    2. Labeling the terms as 'a' and 'b' will make it easier to compare our polynomial to the sum of cubes formula
    3. Our 'a' term is '3x2y'
    1. Our polynomial has been rewritten as: '(3x2y)3+(2)3'
    2. Labeling the terms as 'a' and 'b' will make it easier to compare our polynomial to the sum of cubes formula
    3. Our 'a' term is '3x2y'
    4. Our 'b' term is '2'
    5. "Cubing" is another way of saying "taking something to the 3rd power"
    1. Our polynomial has been rewritten as: '(3x2y)3+(2)3'
    2. Labeling the terms as 'a' and 'b' will make it easier to compare our polynomial to the sum of cubes formula
    3. Our 'a' term is '3x2y'
    4. Our 'b' term is '2'
    1. Our polynomial has been rewritten as: '(3x2y)3+(2)3'
    2. Labeling the terms as 'a' and 'b' will make it easier to compare our polynomial to the sum of cubes formula
    3. Our 'a' term is '3x2y'
    4. Our 'b' term is '2'
    1. Our 'a' term is '3x2y'
    2. Our 'b' term is '2'
    1. Our 'a' term is '3x2y'
    2. Our 'b' term is '2'
    3. 'a' and 'b' are just variables here
    1. Our 'a' term is '3x2y'
    2. Our 'b' term is '2'
    3. 'a' and 'b' are just variables here
    4. Plugging in our values for 'a' and 'b' into the sum of cubes formula will factor our polynomial for us!
    1. The sum of cubes formula is 'a3+b3=(a+b)(a2-ab+2)'
    2. Our 'a' term is '3x2y'
    3. Our 'b' term is '2'
    1. The sum of cubes formula is 'a3+b3=(a+b)(a2-ab+2)'
    2. Our 'a' term is '3x2y'
    3. Our 'b' term is '2'
    4. So: '(a+b)=(3x2y+2)'
    1. The sum of cubes formula is 'a3+b3=(a+b)(a2-ab+2)'
    2. Our 'a' term is '3x2y'
    3. Our 'b' term is '2'
    4. So: '(a+b)=(3x2y+2)'
    5. 'a2=(3x2y)2'
    1. The sum of cubes formula is 'a3+b3=(a+b)(a2-ab+2)'
    2. Our 'a' term is '3x2y'
    3. Our 'b' term is '2'
    4. So: '(a+b)=(3x2y+2)'
    5. 'a2=(3x2y)2'
    6. 'ab=(3x2y)(2)'
    1. The sum of cubes formula is 'a3+b3=(a+b)(a2-ab+2)'
    2. Our 'a' term is '3x2y'
    3. Our 'b' term is '2'
    4. So: '(a+b)=(3x2y+2)'
    5. 'a2=(3x2y)2'
    6. 'ab=(3x2y)(2)'
    7. 'b2=(2)2'
    1. The sum of cubes formula is 'a3+b3=(a+b)(a2-ab+2)'
    2. Our 'a' term is '3x2y'
    3. Our 'b' term is '2'
    4. So: '(a+b)=(3x2y+2)'
    5. 'a2=(3x2y)2'
    6. 'ab=(3x2y)(2)'
    7. 'b2=(2)2'
    1. So we need to simplify: '(3x2y+2)([3x2y2]-3x2y•2+22)'
    1. So we need to simplify: '(3x2y+2)([3x2y2]-3x2y•2+22)'
    2. [3x2y)2]=9x4y2
    3. -3x2y•2=-6x2y
    4. 22=4
    1. We simplified '27x6y3+8' to '(3x2y)(9x4y2-6x2y+4)'