How did you completely factor the sum and difference of two cubes? Write the process of each and their rule of pattern
Answers
Answer:
The other two special factoring formulas you'll need to memorize are very similar to one another; they're the formulas for factoring the sums and the differences of cubes. Here are the two formulas:
Factoring a Sum of Cubes:
a3 + b3 = (a + b)(a2 – ab + b2)
Factoring a Difference of Cubes:
a3 – b3 = (a – b)(a2 + ab + b2)
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The sum or difference of two cubes can be factored into a product of a binomial times a trinomial.
That is, x³+y³= (x+y)(x²−xy+y²)
and
x³−y³=(x−y)(x²+xy+y²) .
In mnemonic for the signs of the factorization is the word "SOAP", the letters stand for "Same sign" as in the middle of the original expression, "Opposite sign", and "Always Positive".
That is, x³±y³=(x[Same sign]y)(x²[Opposite sign]xy[Always Positive]y²)
Example 1:
Factor 27p³+q³ .
Let's write cube of an expression.
27p³+q³=(3p)³+(q)³
Let's use the factorization of sum of cubes to rewrite.
27p³+q³= (3p)³+(q)³
=(3p+q) ((3p)²−3pq+q²)
=(3p+q)(9p²−3pq+q²)
Example 2:
Factor 40a³−625b³ .
Factor out the G CF from the two terms.
40a³−625b³=5(8a³−125b³)
Try to write each of the terms in the binomial as a cube of an expression.
8a³−125b³=(2a)³−(5b)³
Use the factorization of difference of cubes to rewrite.
5(8a³−125b³)=5((2a)³−(5b)³)
=5[(2a−5b)((2a)²+10ab+(5b)²)]
=5(2a−5b)(4a²+10ab+25b²)