College Algebra help?

Question

How did (a3 + b3) = (a + b) (a2 - ab + b2)? Doesn't it equal (a3 –ab + b3)

oh I see. Thanks everybody! =)

wait, to the first answer, why are those rules like that?

Answer 1

a^3 + b^3

This is known as a sum of cubes, and factors as per the following steps:

1) Take the cube root of each term, and place them in one set of brackets.
2) In the second set of brackets, you're going to want to remember the following:
a) Square the first
b) Negative product
c) Square the last.

Let's take this one at a time.

For a^3 + b^3, remember our first step is to take the cube root of each term. Therefore, we have the cube root of a^3 (which is a) and the cube root of b^3 (which is b). Therefore, our first set of brackets contains (a + b).

(a + b) (? + ? + ?)

Now, onto step 2.

"Square the first."

In the first question mark, the value you're going to place in there is the square of the first term in the first step of brackets, hence "square the first." The square of a is a^2, so we have

(a + b)(a^2 + ? + ?)

"Negative product"

What we want to do is multiply both terms in the first set of brackets, and then take the negative. a times b is ab, and taking the negative makes it -ab.

(a + b)(a^2 - ab + ?)

"Square the last"

Self-explanatory. Square the second term in the first set of brackets.

(a + b)(a^2 - ab + b^2)


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As for your second question as to whether it equals
a^3 - ab + b^3, it doesn't; that is

a^3 + b^3 does NOT equal a^3 - ab + b^3.

As a counterexample, let a = 1 and b = 1. Then

a^3 + b^3 = 1^3 + 1^3 = 1 + 1 = 2, and
a^3 - ab + b^3 = 1^3 - (1)(1) + 1^3 = 1 - 1 + 1 = 0 + 1 = 1

Notice that they're not equal. All it takes is one example to show they're not equal.

The reason why the rules are like that is because that's the only way you can get the middle terms to cancel, after multiplying it out.

Let's take (a + b)(a^2 - ab + b^2)

Remember that to multiply this out, we distribute the "a" to each term, and then distribute the b to each term. Let's multiply it out now.

a^3 - (a^2)b + ab^2 + (a^2)b - ab^2 + b^3

Notice how we have -(a^2)b and +(a^2)b, which obliterate each other. The same thing goes for ab^2 and -ab^2, which also cancel each other out. This leaves us with

a^3 - 0 + 0 + b^3, or, just simply

a^3 + b^3

That's the reason why it factors that way; because that's just the way it is.

Answer 2

When you have the problem (a^3 + b^3)
1.) take the cube of each term and use the same sign as the operation in this case you started with + so (a+b)
2.) Square the first term in the first factor (to get the cube back); multiply the two terms in the first factor and use the opposite sign of operation; add the square of the seconde term (a^2 - ab+b^2)

I hope this helps i am taking college algebra right now to and this is how my book says to do it.

Answer 3

Lets just multiply (a + b) (a2 - ab + b2) and see what we get

Using distribution I will change it to
[ a (a2 - ab + b2)] + [ b (a2 - ab + b2) ]

now multiplying

[ a3 - a2b + ab2] + [a2b - ab2 + b3]

now adding and cancelling
a3 + b3

Answer 4

no. distribute and combine like-terms the a2b and the ab2 cancel each other out, leaving you with (a3+b3)

(a+b)(a^2-ab+b^2) =

a3-a2b+ab2+a2b-ab2+b3