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

Algebra ->  Polynomials-and-rational-expressions -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at 4x%5E2-33x%2B65 we can see that the first term is 4x%5E2 and the last term is 65 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 NumberSecond NumberSum
12601+260=261
21302+130=132
4654+65=69
5525+52=57
102610+26=36
132013+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 4x%5E2-33x%2B65, replace -33x with -13x%2B-20x (notice -13x%2B-20x adds up to -33x. So it is equivalent to -33x)

4x%5E2%2Bhighlight%28-13x%2B-20x%29%2B65


Now let's factor 4x%5E2-13x-20x%2B65 by grouping:


%284x%5E2-13x%29%2B%28-20x%2B65%29 Group like terms


x%284x-13%29-5%284x-13%29 Factor out the GCF of x out of the first group. Factor out the GCF of -5 out of the second group


%28x-5%29%284x-13%29 Since we have a common term of 4x-13, we can combine like terms

So 4x%5E2-13x-20x%2B65 factors to %28x-5%29%284x-13%29


So this also means that 4x%5E2-33x%2B65 factors to %28x-5%29%284x-13%29 (since 4x%5E2-33x%2B65 is equivalent to 4x%5E2-13x-20x%2B65)




So 4x%5E2-33x%2B65 factors to %28x-5%29%284x-13%29



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Answer:


So x-5 is a factor of 4x%5E2-33x%2B65