SOLUTION: Use the factor theorem to decide whether or not the second polynomial is a factor of the first. 9) 4x^2 - 33x + 65; x

SOLUTION: Use the factor theorem to decide whether or not the second polynomial is a factor of the first. 9) 4x^2 - 33x + 65; x - 5

Question 127518This question is from textbook College Algebra
: Use the factor theorem to decide whether or not the second polynomial is a factor of the first.
9) 4x^2 - 33x + 65; x - 5 This question is from textbook College Algebra

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Looking at we can see that the first term is and the last term is where the coefficients are 4 and 65 respectively.

Now multiply the first coefficient 4 and the last coefficient 65 to get 260. Now what two numbers multiply to 260 and add to the middle coefficient -33? Let's list all of the factors of 260:

Factors of 260:
1,2,4,5,10,13,20,26,52,65,130,260

-1,-2,-4,-5,-10,-13,-20,-26,-52,-65,-130,-260 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 260
1*260
2*130
4*65
5*52
10*26
13*20
(-1)*(-260)
(-2)*(-130)
(-4)*(-65)
(-5)*(-52)
(-10)*(-26)
(-13)*(-20)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -33? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -33

First Number	Second Number	Sum
1	260	1+260=261
2	130	2+130=132
4	65	4+65=69
5	52	5+52=57
10	26	10+26=36
13	20	13+20=33
-1	-260	-1+(-260)=-261
-2	-130	-2+(-130)=-132
-4	-65	-4+(-65)=-69
-5	-52	-5+(-52)=-57
-10	-26	-10+(-26)=-36
-13	-20	-13+(-20)=-33

From this list we can see that -13 and -20 add up to -33 and multiply to 260

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

So factors to

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Answer:

So is a factor of
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