Solving a cubic function by factoring: using the sum or

difference of two cubes.

By Diane Webb

What is a cube?

1

864

27125

Factoring the sum or difference of two cubes:

(a³+b³) a³ is a perfect cube since a*a*a = a³b³ is a perfect cube since b*b*b = b³

Always remember that the factors of the sum or difference of two cubes is always a (binomial) and a (trinomial).

(a³+b³)=(binomial)(trinomial)

To find the two factors, let’s do the following:

First the binomial: Take the cubed root of each monomial within the problem.l

(a³+b³)=(a+b)(trinomial)

Since, the cubed root of a³ = a and the cubed root of b³ = b.

Now, the trinomial.

The first term of the trinomial isthe first term of the binomial squared.

(a³+b³)=(a+b)(a²+__+__)

The second term of the trinomial is the oppositeof the product of the two terms of the binomial.

(a³+b³)=(a+b)(a²-ab+__)

The third term of the polynomial is the 2nd termof the binomial squared.

(a³+b³)=(a+b)(a²-ab+b²)

The first term of thebinomial is “a” and a*a = a²

The product of “a” and “b” is “ab” and then the opposite tells you to change the sign of the product.

The second term of the binomial is “b” and b*b=b²

Factor (x³-8)Binomial factor is (x-2)Trinomial factor is (x²+2x+4)

Remember that the trinomial is not factorable.

Factored form: (x³-8)=(x-2)(x²+2x+4)

Is it possible to check our answers?

Remember that you may check using either the RemainderTheorem or division.

Remainder theorem: If P(x)=x³-8 and the factor is (x-2),Then P(2)=(2)³-8 = 8 – 8 = 0Since the remainder is 0, then x-2 is a factor. Synthetic division:

2 1 0 0 -8 2 4 8 1 2 4 0This tells you two things: 1. (X-2) is a factor since the remainder is 0. 2. The quotient is x²+2x+4 which is the trinomial factor of the cubic polynomial.

What about 2x³+2?

First of all, you can not forget that theGCF must be factored out of the cubic function. SO, what is the GCF of2x³ and 2?

2 is the GCF. Now, factor the two first: 2(x³+1)Look at the binomial. Is it a differenceor the sum of two cubes. If yes, factor theexpression.2x³+2 = 2(x³+1) = 2(x+1)(x²-x+1)

Now that we can factor the sum and difference of two cubes, let us solve them.

Remember to solve a cubic equation, we need to use our factors. Set yourfactors equal to 0. Then solve for x.

• Go back to the the previous problem: • 2x³+2=0

• Factored form: 2(x+1)(x²-x+1)=0

• Set the factors equal to 0.• 2=0 (x+1)=0 (x²-x+1)=0• 20 so it is not part of our solution.• X+1=0 so x = -1.• What about x²-x+1=0? Remember we talked

about the fact that it is not factorable. How do we solve that quadratic?

• QUADRATIC FORMULA!!