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Chapter 2: Problem 48

Verify that \(x^{3}+y^{3}=(x+y)\left(x^{2}-x y+y^{2}\right)\).

Short Answer

Expert verified
We verified the given equation by expanding the right side \((x+y)(x^2-xy+y^2)\) using the distributive property and then combining like terms. After simplifying, we found that the right side of the equation equals to \(x^3 + y^3\), which matches the left side. Therefore, the given equation is true: \[x^3 + y^3 = (x+y)(x^2 - xy + y^2)\].

Step by step solution

01

Expand the right side expression

First, we'll expand the expression on the right side which is \((x+y)(x^2-xy+y^2)\). We can use the distributive property (also known as the FOIL method) to multiply these two binomials. \[ (x+y)(x^2-xy+y^2) = x \cdot (x^2 - xy + y^2) + y \cdot (x^2 - xy + y^2)\]
02

Distribute x and y terms

Now, distribute the x and y terms to each term within the parentheses. \[x(x^2 - xy + y^2) = x^3 - x^2y + xy^2\] \[y(x^2 - xy + y^2) = yx^2 - y^2x + y^3\]
03

Combine the distributed terms

Now, combine the distributed terms from the previous step to simplify the expression further. \[x^3 -x^2y + xy^2 + yx^2 - y^2x + y^3\]
04

Rearrange terms

Rearrange the terms in alphabetic order and sum the like terms. \[x^3 + x^2y - x^2y + xy^2 + y^3 + yx^2 - y^2x\] The above equation now simplifies to: \[x^3 + y^3\]
05

Compare the simplified right side to the left side

Now that we have simplified the right side of the equation, we can see that it is equal to the left side of the equation: \[x^3 + y^3 = x^3 + y^3\] Since the simplified right side is equal to the left side, we have verified that the given equation is true: \[x^3 + y^3 = (x+y)(x^2 - xy + y^2)\]

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Most popular questions from this chapter

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

Suppose \(q\) is a polynomial of degree 5 such that \(q(1)=-3\). Define \(p\) by $$ p(x)=x^{6}+q(x) $$ Explain why \(p\) has at least two zeros.

Find all real numbers \(x\) such that $$ x^{4}+5 x^{2}-14=0 $$.

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

Explain why the composition of two rational functions is a rational function.
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