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Chapter 6: Problem 76

Factor by grouping. $$x^{3}+6 x^{2}-2 x-12$$

Short Answer

Expert verified
The factored form of the expression \(x^{3}+6 x^{2}-2 x-12\) by grouping is \((x+6)(x^2-2)\).

Step by step solution

01

Group the terms

Group the given expression \(x^{3}+6 x^{2}-2 x-12\) into two groups as \(x^{3}+6 x^{2}\) and \(-2 x-12\).
02

Factor out the common factor from each group

Factor out the common factor in each group: \(x^2\) from the first group and \(-2\) from the second group. This results in \(x^2(x+6)-2(x+6)\)
03

Factor by grouping

In the expression \(x^2(x+6)-2(x+6)\), notice the common binomial factor of \((x+6)\) in both terms. We can factor out this common binomial term to get \((x+6)(x^2-2)\).

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Most popular questions from this chapter

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 \(7250 ?\)

Solve equation and check your solutions. \(x^{3}-4 x=0\)

In Exercises \(130-133,\) use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{3}-1=(x-1)\left(x^{2}-x+1\right)$$

Solve each equation. \(x^{3}-x^{2}-16 x+16=0\)

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