Surely Simple (What am I Overlooking?)! (x-2)^5 + 5(x+2)(x-2)^4 becomes (x-2)^4(6x+8)

[limaecho1988]

I am teaching my self and have obviously missed something.

How would you express the following in the terms given in the second part? I am using text book which gives answers in the latter form.

Examples:

1. (x-2)^5 + 5(x+2)(x-2)^4 becomes (x-2)^4(6x+8)

2. 4(x+1)^2(x-1)^3 + 2(x+1)(x-1)^4 becomes 2(x+1)(x-1)^3(3x+1)

Thanks in advance.

Also it would be nice to know how to write formula style on here.!

 

[JeffM]

I am teaching my self and have obviously missed something.

How would you express the following in the terms given in the second part? I am using text book which gives answers in the latter form.

Examples:

1. (x-2)^5 + 5(x+2)(x-2)^4 becomes (x-2)^4(6x+8)

2. 4(x+1)^2(x-1)^3 + 2(x+1)(x-1)^4 becomes 2(x+1)(x-1)^3(3x+1)

Thanks in advance.

Also it would be nice to know how to write formula style on here.!

It is a little confusing because the two examples are inconsistent. But both are simply exercises in simplification through factoring.

In the first, you factor out \(\displaystyle (x - 2)^4.\)

\(\displaystyle (x - 2)^5 + 5(x + 2)(x - 2)^4 = (x - 2)(x - 2)^4 + 5(x + 2)(x - 2)^4 = \)

\(\displaystyle (x - 2)^4\{(x - 2) + 5(x + 2)\} = (x - 2)^4(x - 2 + 5x + 10) =\)

\(\displaystyle (x - 2)^4(6x + 8) = 2(x - 2)^4(3x + 4).\)

In the second, you factor out \(\displaystyle (x + 1)(x - 1)^3.\)

\(\displaystyle 4(x + 1)^2(x - 1)^3 + 2(x + 1)(x - 1)^4 =\)

\(\displaystyle (x + 1)(x - 1)^3(4)(x + 1) + (x + 1)(x - 1)^3(2)(x - 1) =\)

\(\displaystyle (x + 1)(x - 1)^3\{4(x + 1) + 2(x - 1)\} =\)

\(\displaystyle (x + 1)(x - 1)^3(4x + 4 + 2x - 2) = \)

\(\displaystyle (x + 1)(x - 1)^3(6x + 2) = 2(x + 1)(x - 1)^3(3x + 1).\)