factoring polynomials

Question

please help factoring these problems

 

 

1. a^2 + a^2    (I think the answer is a^2(a^2 + 1), but I don't know how to explain my steps.

 

2.  8x^3 - 2x  (I have 2x(2x + 1)(2x - 1)

 

 

How do I factor  8a^3 - 27

 

thanks

Answer 1

As to your third one, that's the difference of 2 cubes.  You'll want to start recognizing some cubes, i.e. 8 is 2³, 27 is 3³, 64 is 4³... probably enough for now.  (And don't forget that 1 is also 1³.)
 
a³ - b³ = (a - b)(a² + ab + b²)
 
Part of the trick here is working with both the numbers and the variables.
 
It will be easier to make yours 8x³ - 27 so as not to mix up a and the variable.  "a" is actually 8x, not just the variable x.  That's why I changed it to x.  So the "a" consists of both the 8 and the x, both coefficient and variable.
 
I would start by getting your cubes of everything: 2, x and 3.  Every time you have a, that's 2x.  Every time you have b, that's 3.
 
Now let's deal with a² and b².  If you factor out 2x from 8x³, what do you have left?  The squares, hence a².  If you factor out 3 from 27, what do you have left?  The square of it, hence b².

Just see what you can do with that.  I hope you have done easier ones than this first, like x³ - 27, since this one is a bit more difficult.  It might be easier to do that first, and then go back and deal with the 8 separately, remembering that's part of a and would go wherever a is.

Answer 2

L - 

Let's look at these one by one. My favorite way to test whether or not how I factored something is correct is by plugging in a value for my variable (a, x, Θ, etc.) and see if my factored expression matches the original one. For #1, let's try that with a=3. 

 

Original expression:  a² + a²

if a=3:                     3² + 3²

calculate exponents: 9 + 9

add:                        18

 

So that's only true if a=3, and when we factor, we don't want to change the value of the expression, so if you factored it correctly, you'd get the same answer in your expression when a=3.

 

Your expression:      a²(a² + 1)

if a=3:                    3²(3² + 1)

calculate exponents: 9(9 + 1)

add parentheses:     9(10)

multiply:                  90

 

Hm, so it looks like your answer cannot be the correct one, because for the same value of a, the expressions have different values. One important thing to remember with expressions and equations is that you can combine like terms, meaning 3x + 1x = 4x, or 2y² - 6y² = -4y². Let's see what that looks like with your first expression. Adding coefficients of 1 (implied when there's no coefficient) can help show that!

 

original expression:  a² + a²

add coefficients:      1a² + 1a²

combine like terms: 2a²

 

Ok, so now we have a single term. Let's plug in a=3  to see if we get the same answer as the original expression.  

 

 

simplified expression: 2a²

a=3:                         2(3²)

calculate exponent:    2(9)

multiply:                    18

 

If you need to pull it apart (factor) it further, it consists of 3 parts multiplied together: 2, a, and a, or 2(a)(a).

 

Let's move on to number 2. You got this one right on your own! Good job! This is how I got the same answer, step by step. 

 

Original expression:          8x³ - 2x

Pull out common multiple: x(8x² - 2)

Pull out common multiple: 2x(4x² - 1)

Pull out common multiple: 2x(2x+1)(2x-1)

 

We can test this again by giving x a value. Let's try x=4

 

Original expression: 8x³ -  2x

x=4:                       8(4)³ - 2(4)

calculate exponent:  8(64) - 2(4)

multiply each term:  512 - 8

subtract:                 504

 

Factored expression: 2x(2x+1)(2x-1)

x=4:                        2(4)[2(4)+1][2(4)-1]

multiply:                   8(8+1)(8-1)

add/subtract:            8(9)(7)

multiply:                   504

 

Awesome! We got it!

I hope this was helpful, L!

Amanda