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Chapter 2: Problem 20

Factor \(x^{16}-y^{8}\) as nicely as possible.

Short Answer

Expert verified
The factored expression for \(x^{16} - y^8\) is \((x^8 + y^4)(x^4 + y^2)(x^2 + y)(x^4 + y^2)(x^2 + y)(x^2 - y)\).

Step by step solution

01

Identify Difference of Squares Pattern and First Iteration-

Note that the exponents in the expression \(x^{16} - y^8\) are even, allowing us to use the difference of squares formula. Applying the formula, we have: \[ x^{16} - y^8 = (x^8 + y^4)(x^8 - y^4) \]
02

Factor the Second Iteration-

Now that the first iteration has been factored, we can once again apply the difference of squares formula to the term \((x^8 - y^4)\): \[ x^8 - y^4 = (x^4 + y^2)(x^4 - y^2) \]
03

Factor the Third Iteration-

Since the difference of squares pattern is still present in the term \((x^4 - y^2)\), we make one last application of the formula: \[ x^4 - y^2 = (x^2 + y)(x^2 - y) \]
04

Combine All Factored Terms-

Now that we have factored all the iterations, we combine the results from Steps 1 to 3 to obtain the final factored expression: \[ x^{16} - y^8 = (x^8 + y^4)(x^4 + y^2)(x^2 + y)(x^8 - y^4) \] And from step 2 and 3, we already have the factors for \((x^8 - y^4)\). So we can substitute them: \[ x^{16} - y^8 = (x^8 + y^4)(x^4 + y^2)(x^2 + y)((x^4 + y^2)(x^4 - y^2)) \] And then substitute once more from step 3: \[ x^{16} - y^8 = (x^8 + y^4)(x^4 + y^2)(x^2 + y)((x^4 + y^2)((x^2 + y)(x^2 - y))) \]
05

Final Answer-

Now that all the factored terms have been combined, we have the final factored expression: \[ x^{16} - y^8 = (x^8 + y^4)(x^4 + y^2)(x^2 + y)(x^4 + y^2)(x^2 + y)(x^2 - y) \]

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Most popular questions from this chapter

Write the domain of the given function \(r\) as a union of intervals. $$ r(x)=\frac{5 x^{3}-12 x^{2}+13}{x^{2}-7} $$

Give an example of a polynomial \(p\) of degree 6 such that \(p(0)=5\) and \(p(x) \geq 5\) for all real numbers \(\mathcal{X}\).

Show that $$ (a+b)^{3}=a^{3}+b^{3} $$ if and only if \(a=0\) or \(b=0\) or \(a=-b\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r \circ t)(x) $$

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r s)(x) $$
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