Expression Equivalency

Question

(ax + 3)(5x^2 - bx + 4) = 20x^3 - 9x^2 - 2x + 12

The equation above is true for all x, where "a" and "b" are constants. What is the value of ab?

Answer 1

Don't waste time doing the whole distribution. Just put the x^2 terms equal to each other and solve for ab.

-abx^2 + 15x^2 = -9x^2

-ab = -9 -15

ab = 24

Answer 2

Multiply ax by all terms in the second (), and then multiply 3 by all terms in the second ().

(ax + 3)(5x^2 - bx + 4) = 5ax^3 - abx^2 + 4ax + 15x^2 -3bx + 12 =

5ax^3 + (15 - ab)x^2 + (4a-3b)x + 12 = 20x^3 - 9x^2 - 2x + 12

5a = 20

a = 20/5 = 4

15-ab = -9

15 - 4b = -9

-4b = -24

b = -24/-4 = 6

ab = 4*6 = 24

You can check with the term in front of x: 4a - 3b = -2

4*4 - 3*6 = -2

16 - 18 = -2

-2 = -2 YES