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Chapter 6: Problem 74

Factor by grouping. $$x^{3}-3 x^{2}+4 x-12$$

Short Answer

Expert verified
The factored form of the cubic polynomial \(x^{3} - 3x^{2} + 4x - 12\) is \((x - 3)(x^{2} + 4)\).

Step by step solution

01

Grouping Terms

Rewrite the expression in four terms as two groups of two terms: \(x^{3}-3 x^{2}\) and \(+4 x-12\)
02

Factoring Each Group

Factor out the greatest common factor from each group. For the first group \(x^{3}-3 x^{2}\), we can factor out \(x^2\), which leaves \(x - 3\). For the second group \(4 x - 12\), we can factor out \(4\), which leaves \(x - 3\). So the expression becomes \(x^{2}(x-3) + 4(x-3)\).
03

Factoring Using Grouping

Now, we observe that our expression is the sum of two terms, both of which include the factor \(x - 3\). We can factor out this common factor to give the final factored form. So the factored form of our expression is \((x - 3)(x^{2} + 4)\).

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Most popular questions from this chapter

In Exercises \(137-141,\) factor completely. $$(x+5)^{2}-20(x+5)+100$$

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(4 x(x+1)=15\)

The alligator, at one time an endangered species, is the subject of a protection program. The formula $$P=-10 x^{2}+475 x+3500$$ models the alligator population, \(P,\) after \(x\) years of the protection program, where \(0 \leq x \leq 12 .\) Use the formula to solve. After how long is the population up to \(5990 ?\)

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In Exercises \(137-141,\) factor completely. $$5 y^{5}-5 y^{4}-20 y^{3}+20 y^{2}$$
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