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Chapter 6: Problem 73

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

Short Answer

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

Step by step solution

01

Group Terms

First, let's rewrite the polynomial \(x^{3} - 2x^{2} +5x - 10\) and isolate each pair of terms: \((x^{3} - 2x^{2}) + (5x - 10)\).
02

Factor Out GCF from Each Group

Now, let's factor out the GCF from each pair. From \(x^{3} - 2x^{2}\), we can factor out \(x^{2}\), resulting in \(x^{2}(x - 2)\). From \(5x - 10\), we can factor out 5, resulting in \(5(x - 2)\). After doing so, the equation becomes \(x^{2}(x - 2) + 5(x - 2)\).
03

Factor the Resulting Expression

The resulting expression, \(x^{2}(x - 2) + 5(x - 2)\), has a common factor of \(x - 2\). We factor this out to obtain \((x - 2)(x^{2} + 5)\), which is the factored form of the original polynomial.

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Most popular questions from this chapter

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$b x^{2}-4 b+a x^{2}-4 a$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$3 x^{2}+243$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$100 y^{2}-49$$

The polynomial \(4 x^{2}+100\) is the sum of two squares and therefore cannot be factored. If the general factoring strategy is used to factor a polynomial, at least two factorizations are necessary before the given polynomial is factored completely.

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$10 x^{4}+20 x^{3}+15 x^{2}$$
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