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Proving Polynomial Identities

Proving Polynomial Identities 1.9 - Solution

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a
The difference of squares has the form (a+b)(ab)=a2b2 (a + b)(a - b) = a^2 - b^2 and we see that our expression has the same form, with a=x2a=x^2 and b=12b = 12.
(x212)(x2+12)\left(x^2 - 12\right)\left(x^2 + 12\right)
(x2)2122\left(x^2\right)^2 - 12^2
(x2)2144\left(x^2\right)^2 - 144
x4144x^4 - 144
b
We'll proceed in the same way as before and rewrite the expression using the difference of squares. Remember that the whole of the term 2y2y, including the 22, must be squared.
(23+2y)(232y)\left(2^3+2y\right)\left(2^3 - 2y\right)
(8+2y)(82y)\left(8+2y\right)\left(8 - 2y\right)
82(2y)28^2 - \left(2y\right)^2
8222y28^2 - 2^2y^2
644y264 - 4y^2
c
Here, the terms are not placed in the same order in the parentheses. However, if we start with simplifying the first parentheses and then change the order of the terms in (-a+2)(\text{-} a + 2), we can use the difference of squares here as well.
(2(-a))(-a+2)(2 -(\text{-} a))(\text{-} a + 2)
(2+a)(-a+2)(2+a)(\text{-} a + 2)
(2+a)(2a)(2+a)(2-a)
22a22^2 - a^2
4a24 - a^2