SOLUTION: Find ax3 + bx2+ cx + d so that a,b,c, and d are integers. a/c =b/d = 4 and a/b =7/5. Factor the result by grouping.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find ax3 + bx2+ cx + d so that a,b,c, and d are integers. a/c =b/d = 4 and a/b =7/5. Factor the result by grouping.       Log On


   

Question 655556: Find ax3 + bx2+ cx + d so that a,b,c, and d are integers.
a/c =b/d = 4 and a/b =7/5. Factor the result by grouping.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
From a/c = 4 we get
a = 4c
which tells us that a is a multiple of 4

From b/d = 4 we get
b = 4d
which tells us that b is a multiple of 4

From a/b = 7/5 we get
5a = 7b
which tells us that a is a multiple of 7 and b is a multiple of 5.

Since b must be a multiple of 5 and 4, we can use 20 for b. With b = 20 and b = 4d, we find that d will be 5. With b = 20 and 5a = 7b we can find that a = 28. And with a = 28 and a = 4c we get c = 7.

So our polynomial is:
28x%5E3%2B20x%5E2%2B7x%2B5
To factor this by grouping we start by grouping:
%2828x%5E3%2B20x%5E2%29%2B%287x%2B5%29
Then we factor out the greatest common factor (GCF) of each group. The GCF of the first group is 4X%5E2. The GCF of the second group is just 1. (Factoring by grouping is one of the rare times we bother to factor out a 1.) Factoring out each GCF from each group:
4x%5E2%2A%287x%2B5%29%2B1%2A%287x%2B5%29
As we can see, the two sub-expressions have a common factor: 7x+5. Factoring out the common factor from each sub-expression:
%287x%2B5%29%2A%284x%5E2%2B1%29

Neither of these factors will factor further so we are finished factoring.