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Chapter 6: Problem 71

Factor by grouping. $$x^{2}+3 x-5 x-15$$

Short Answer

Expert verified
The factorization of the polynomial \(x^{2}+3x-5x-15\) by grouping is \(x(x - 2) - 15\).

Step by step solution

01

Combine Like Terms

First, combine the terms in the polynomial that are similar, to simplify the equation. The polynomial \(x^{2}+3x-5x-15\) simplifies to \(x^{2} - 2x - 15\).
02

Group the Terms

The next step is to group the terms that have common factors. In this case, it's best to group the \(x^{2}-2x\) together and \(-15\) separately. The grouped polynomial then becomes \((x^{2}-2x) - 15\).
03

Factor the Groups

Factor out the common factor from each group. From the first group, \(x\) can be factored out as a common factor, leaving \(x(x-2)\). The second group is already as simplified as possible. So, the polynomial is now \(x(x - 2) - 15\).
04

Final Expression

Notice that the polynomial now is in the factored form, and we cannot further simplify it. So the final expression is \(x(x - 2) - 15\).

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Most popular questions from this chapter

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. $$2 r^{3}+30 r^{2}-68 r$$

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. $$3 r^{3}-27 r^{2}-210 r$$

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. $$2 y^{3}+3 y^{2}-50 y-75$$

You are about to take a great picture of fog rolling into San Francisco from the middle of the Golden Gate Bridge, 400 feet above the water. Whoops! You accidently lean too far over the safety rail and drop your camera. The height, in feet, of the camera after \(t\) seconds is modeled by the polynomial \(400-16 t^{2} .\) The factored form of the polynomial is \(16(5+t)(5-t) .\) Describe something about your falling camera that is easier to see from the factored form, \(16(5+t)(5-t),\) than from the form \(400-16 t^{2}\)
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