Find two different values that complete each expression so that the trinomial can be factored into the product of two binomials. Factor your trinomial

Question

15v^2-?v-24

Answer 1

One way to figure this out is to realize that when you are factoring ax² + bx + c over the set of integers, that the value of b depends on the multiples of the product of a and c. If c is negative then b is the difference of two multiples; if c is positive then b is the sum of two multiples.

 

In your problem the product of a and c is 15(-24)=-360. So find two multiples of -360. This is easier to do if you write with prime factorization: 2(2)(2)(3)(3)(5).

 

So one set of multiples is 8 times 45. Since we want a difference and we want the b to be negative, then let's use -45x and +8x to write the polynomial like this: 15x² -45x + 8x - 24. Now this can be factored by grouping:

 

(15x² - 45x) + (8x - 24) = 15x(x-3) + 8(x-3) = (x-3)(15x+8)

 

So the value of the ? for this particular instance is -37. You should be able to follow the above pattern for any two multiples of -360.

Answer 2

15v² - 9v - 24 = (15v - 24)(v + 1) is one way to factor.

 

Another: (15v + 24)(v - 1) = 15v² + 9v -24  where the ? mark is followed by -1, thus making the middle coefficient positive.