polymials factor

Question

 

15x²+26x+7

Answer 1

You can use AC factoring when the first coefficient is not = 1.

It's similar to simple factoring, but with a little more work in the middle.

 

ax² + bx + c

Find two factors of the ac product that add up to b.

15x² + 26x + 7

ac = (15)(7) = 105

Find the factors of 105. Some may repeat.  24 would factor into (2)(2)(2)(3).

3 |_105

5 |__35

7 |___7

         1

 

You need to use all of the factors exactly once.  Try splitting them into two groups until you find the right combination.

f₁ = (3)(5) = 15

f₂ = (7)(1) = 7

f₁ + f₂ = 22  Too small.

 

f₁ = (3)(7) = 21
f₁ = (5)(1) = 5
f₁ + f₂ = 26    That's it.

 

21 and 5 are the factors we need.

 

Split the middle term (26x) into 21x + 5x.  (5x + 21x would work just as well!)

15x² + 21x + 5x + 7

Group the terms and simplify the groups.

(15x² + 21x) + (5x + 7)

3x(5x + 7) +1(5x + 7)

 

Factor out the (5x + 7)

(5x + 7)(3x+1)

 

Done

 

Answer 2

 

15x²+26x+7

 

= (3x + 1)(5x + 7)

 

When I factor problems like this, I usually factor the lead coefficient and try different combinations.  In this case the factors are 1 & 15, or 3 & 5 so it was pretty clear which one would work.

 

Hope this helps.