Answer:
12) Yes, it is a factor of p(x); (x - 1) and (x + 1)
13) No, it is not a factor of p(x); p(x) is not factorable
14) Yes, it is a factor of p(x); (x - 5) and (x - 9)
Explanation:
12) p(x) = x³ + 2x² - x - 2
We want to know if x+2 is a factor of the above polynomial. So, we can start by factoring p(x). We can factor by grouping here.
p(x) = x³ + 2x² - x - 2
= x²(x + 2) - (x + 2)
= (x² - 1)(x + 2)
So x+2 is a factor of p(x). We're going to find the remaining factors because x² - 1 can be factored further.
= (x - 1)(x + 1)(x + 2)
To answer the question:
So (x + 2) IS a factor of p(x) and the remaining factors are (x - 1) and (x + 1).
13) p(x) = 2x⁴ + 6x³ - 5x - 10
For this one, I can see that factoring by grouping won't work, so I'll try the Factor Theorem. We want to know is (x + 2) is a factor of p(x), so we will set x equal to -2.
-2| 2 6 -5 -10
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I've written the x-value in the box in the corner there, and I've listed the coefficients of the terms of the polynomial in order.
I'm going to bring the 2 straight down under the bar. Multiply the number in box by whatever is brought down. In this case it is -2 * 2 so that is -4. Put this number directly below the number in the next column and add, putting that sum under the bar. Multiply by the number in the box, move to the next column, add, bring down, repeat. This sounds complicated but I trust that once you see it, you'll recognize the pattern.
-2| 2 6 -5 -10
↓ -4 -4 18
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2 2 -9 8
After there are no more columns left to work, look at the last sum calculated. If it is equal to 0, than the polynomial we were looking at IS a factor of p(x). If not, then (x + 2) is not a factor. Because 8 ≠ 0, (x + 2) is not a factor of p(x), nor is this binomial factorable.
14) p(x) = x³ - 22x² + 157x - 360
We want to see if (x - 8) is a factor of p(x). I can tell that this binomial will be difficult to factor by grouping, so I'll use the Factor Theorem. This time our factor is (x - 8), so I will set x equal to 8, such that the factor becomes equal to 0. Repeat the process shown above, and if you still can't grasp it, look up "how to use the Factor Theorem."
8| 1 -22 157 -360
↓ 8 -112 360
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1 -14 45 0
So from this, we can say that (x - 8) is a factor of p(x). Simply using this factor theorem, we can see that this bottom line represents the coefficients for p(x) / (x - 8)
p(x) = (x² - 14x + 45)(x - 8)
See the coefficients from the line above? Pretty cool. So now you can factor this last polynomial like you would any other. The last coefficient is positive and the middle coefficient is negative, meaning both factors are negative numbers. -5 and -9 sum to -14; -5 * -9 = 45;
p(x) = (x - 5)(x - 9)(x - 8)
So (x - 8) IS a factor of p(x) and the remaining factors are (x - 5) and (x - 9).