Factor the following completely

Question

-8x^4 + 2x^2

 

x^2 - 2x - 35

 

3x^3 + 9x^2 - 54x

 

x^3 - 4x

Answer 1

Hi Sean from Bay City, TX. Hope you are having a great day at school.

A great Algebra polynomial expression to factor. Every student's favorite?

Right??  NOT!

Well, I'm glad you reached out to Wyzant for some assistance.

Now, I'm going to help you with developing a system on answer these, however, I won't just give you the answer, that would be too easy. Plus, if you have to do this on your own, like in a test, you'd be lost. 

 

                                 -8x⁴ + 2x²       

the first thing we need to make sure of is, do you know what the terms coefficients and variable mean. They help you understand what it is you are being asked to do: If you don't you need to make sure you do.

             let's look at the first expression: -8x  and then compare it with 

             the second expression. What do they have in common?

             Well 8 and 2 are multiples of 2 AND they have the same

              variable x. Now there is a (-) in from of the 8. What can you do

             to change it. Let's rewrite the expression:

                  (-1)(8x⁴)   +  (2x²)  again rewrite the expression:

    

                  (-1)[8x⁴  -  2x²]   do you see what you did? Rewrite it again

      Since we moved the negative sign on the outside, we change the plus
       sign to an negative sign, Check it:
       multiply (-1)(8x4) and (-1)(2x2) and put it into the expression

                                -8x⁴ -  -2x²

        Since the negative sign is next to the subtraction sign, you can 

          change that to a plus sign:

                            -8x⁴ + 2x²    which is what your original expression is

     Make sure you understand what is being done.  Let's continue:

        (-1)[(2•4)x⁴ - (2•1)x²]   Do you see what you did? Rewrite it again

          (-1•2)[(4)x⁴ - (1)x²]  Move the 2 so you will rewrite the expression

                 -2(4x⁴  -  x²)   do you see what you did?

        to check to make sure this is correct complete the multiplication

(-2)(4x⁴) - (-2)(x²) ⇒ which gives you  -8x⁴ - -2x²  same as  -8x⁴ + 2x² 

            So far you have facted the -8x⁴ + 2x²   ⇒  (-2)(4x⁴  - x²)

  There are several more steps...factors you can make for   (4x⁴ - x²)

           Whenever you see something similar to this:   xⁿ  -  y^m   

           it can be factored into something like this:  (xⁿ + y^m)(xⁿ - y^m)

   I'll do this for you, but practice this and you will get the hang of it:

          for 4x⁴  -  x²    it can be factored to:   (2x² + x)(2x² - x)

      Have you learned and practiced FOIL: Front, Outer,Inner,Last

                 let's do that, here goes:

              Front:  (2x²)(2x²)  ⇒   4x⁴

              Outer:  (2x²)(-x)   ⇒   -2x³

              Inner:   (x)(2x²)    ⇒   2x³

              Last:     (-x)(x)      ⇒   -x²

Now line them up and add

     4x⁴  +  -2x³  +  2x³  + -x²   ⇒   4x⁴ -  2x³ + 2x³ -  x²  ⇒  4x⁴ - x²

So your final factor for         -8x⁴ + 2x² 

                     -2[(2x² + x)(2x² - x)]  which is equal to    -8x⁴ + x²

Hope this helps. Do each of the rest just like this and you will be a pro

Contact me on Wyzant if you need any further assistance.