b Examining the expression, we notice that 8x^3 and 2x^7 contain common factors. Therefore, we should start by breaking them down into factors until we identify these common factors.
2* 4 * x* x * x_(8x^3) - 2* x* x* x * x^4_(2x^7)The terms share one 2 and three x. Therefore, the greatest possible factor we can pull out from both of them is 2x^3
8x^3-2x^7 ⇔ 2x^3( 4-x^4 )
Looking at the parentheses, we see that 4 is a perfect square and so is x^4 if we rewrite it as a power of a power. Therefore, like in Part A, we can further factor this by rewriting the expression in the parentheses as a difference of squares.
d If we compare this to Part B, we see that the only thing that is different is the operation between the terms.
8x^3 - 2x^7
8x^3 + 2x^7
In Part B, we identified the greatest common factor between 8x^3 and 2x^7 to be 2x^3. We can still factor our expression by pulling out this greatest common factor but that is how far we can go.
8x^3+2x^7 ⇔ 2x^3 (4+x^4 )
e Whenever we can write an expression as a difference of squares it can be factored to a product of a conjugate pair of binomials.
