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Solving a cubic function by factoring: using the sum or difference of two cubes. By Diane Webb. What is a cube?. 27. 1. 125. 8. 64. Factoring the sum or difference of two cubes:. (a ³+b³). a ³ is a perfect cube since a*a*a = a³. b ³ is a perfect cube since b*b*b = b ³. - PowerPoint PPT Presentation
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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!!