Question

Hi, I've got the Barron's ACT Flashcard Set and one of the questions seems a little off:
Multiple-Choice 76
If (3x + a) · (x + b) = 3x^2 - 19x - 14, what are a and b?
A. a=1, b=19
B. 1=2, b=7
C. a=-1, b=14
D. a=2, b=-7
E. a=1, b=-14
I did the FOIL, but it doesn't get rid of x all the way, so I can't figure out how to get a and b.

Thank you!

Answer 1

first expand (3x + a)(x + b)
= 3x^2 + 3bx + ax + ab
= 3x^2 + x(3b + a) + ab

Answer 2

now compare coefficients with
3x^2 - 19x - 14 = 0

Answer 3

this gives
3b + a = -19
ab = -14

solve this system of equations

Answer 4

* ignore the '= 0' call it a typo!

Answer 5

What does it mean to compare coefficients? Wouldn't that be to set them equal to each other?

Answer 6

yes the coefficient of x in one expression is 3a+b and the other its -19
so we can set them equal

Answer 7

i should have said equate coefficents

Answer 8

So wouldn't that give us 3x^2 + x(3b + a) + ab = 3x^2 - 19x - 14 ?

Answer 9

right - though there should be an identity sign not an equal sign in between

Answer 10

an identity differs from an equation in that an identity is true for all values of the variable
as its an identity we can equate coefficients

Answer 11

Strictly speaking the original equation should have used the identity sign not the '='.