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Chapter 0: Problem 15

Find each product. $$(x+1)\left(x^{2}-x+1\right)$$

Short Answer

Expert verified
The product of \((x+1)\) and \(\left(x^{2}-x+1\right)\) is \(x^{3}+1\).

Step by step solution

01

Distribute the first term in the first polynomial

Distribute 'x' from \((x+1)\) to all terms in \(\left(x^{2}-x+1\right)\). Thus, we get \(x\cdot x^{2}, x \cdot -x\), and \(x \cdot 1\), which simplifies to \(x^3, -x^2\), and \(x\), respectively.
02

Distribute the second term in the first polynomial

Next, distribute '+1' from \((x+1)\) to all terms in \(\left(x^{2}-x+1\right)\). We get \(1 \cdot x^{2}\), \(1 \cdot -x\), and \(1 \cdot 1\), which simplifies to \(x^2, -x\), and \(1\), respectively.
03

Combine all the results

After distributing, combine all the terms to find the final product. That gives us \(x^{3}-x^{2}+x +x^{2}-x+1\).
04

Simplify the Equation

On simplifying, any like terms in the equation will be added together, giving us \(x^{3}+1\).

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Most popular questions from this chapter

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49$$

Rationalize the numerator. $$\frac{\sqrt{x}+\sqrt{y}}{x^{2}-y^{2}}$$

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$8^{-\frac{1}{4}}=-2$$

Perform the indicated operations. Simplify the result, if possible. $$\left(2-\frac{6}{x+1}\right)\left(1+\frac{3}{x-2}\right)$$

Perform the indicated operations. Simplify the result, if possible. $$\frac{y^{-1}-(y+5)^{-1}}{5}$$
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