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Chapter 0: Problem 11

Factor by grouping. $$x^{3}-2 x^{2}+5 x-10$$

Short Answer

Expert verified
The factored form of the given polynomial \(x^{3}-2 x^{2}+5 x-10\) is \((x-2)(x^{2} + 5)\).

Step by step solution

01

Arrange the polynomial

The given polynomial is \(x^{3}-2 x^{2}+5 x-10\). Here, the terms do not need to be rearranged since they are in standard form (highest degree to lowest degree).
02

Group the terms

Next, group the terms into two pairs. Here, the grouping will look like this: \((x^{3}-2 x^{2}) + (5 x-10)\).
03

Factor out the common factors

In the first group, \(x^{2}\) is a common factor and in the second group, \(5\) is a common factor. Now, factoring out these common factors, we get \(x^{2}(x-2) + 5(x-2)\).
04

Factor the grouped polynomial

Now, notice that \((x-2)\) is a common factor in both terms. So, we can factor out \((x-2)\), resulting in \((x-2)(x^{2} + 5)\). This is the factored form of the given polynomial.

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Most popular questions from this chapter

Using an example, explain how to factor out the greatest common factor of a polynomial.

Will help you prepare for the material covered in the next section. A. Use a calculator to approximate \(\sqrt{300}\) to two decimal places. B. Use a calculator to approximate \(10 \sqrt{3}\) to two decimal places. C. Based on your answers to parts (a) and (b), what can you conclude?

Factor and simplify each algebraic expression. $$12 x^{-\frac{3}{4}}+6 x^{\frac{1}{4}}$$

In parts (a) and (b), complete each statement. $$\text { a. } b^{4} \cdot b^{3}=(b \cdot b \cdot b \cdot b)(b \cdot b \cdot b)=b^{?}$$ $$\text { b. } b^{5} \cdot b^{5}=(b \cdot b \cdot b \cdot b \cdot b)(b \cdot b \cdot b \cdot b \cdot b)=b^{7}$$ c. Generalizing from parts (a) and (b), what should be done with the exponents when multiplying exponential expressions with the same base?

Factor completely. $$2 x^{2}-7 x y^{2}+3 y^{4}$$
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