Question

WILL GIVE MEDAL! Please explain!

52. Given that (x+2) and (x-1) are factors of the quadratic expression below, what are the values of a and b?
x^2 + (a+2)x + a + b

a b
F. -4 5
G. -3 1
H. -3 5
J. -1 3
K. -1 -1

Answer 1

\[x^2 + (a+2)x + a + b\]

Answer 2

What do you get if you multiply \((x+2)(x-1)\)

Answer 3

\[x^2+x-2\]

Answer 4

Okay, so let's mix and match.

\[x^2 + (a+2)x + a + b\]\[x^2 + x -2\]

What do \(a,b\) have to be so that \((a+2)x = x\) and \(a+b = -2\)?

Answer 5

I haven't a clue...

Answer 6

Oh, come on.

\[(a+2) x = x\]Solve that for \(a\)

Answer 7

ax + 2x = x
ax = x - 2x
a = 1 - 2

Answer 8

so \(a =\)

Answer 9

-1

Answer 10

right.

now how about \(a+b =-2\) given that \(a = -1\)?

Answer 11

b = 1

Answer 12

OH NOW I GET IT!!!!

Answer 13

Because x^2 - sum x + product = 0

Answer 14

that's also a good way of looking at it, yes

Answer 15

One of those problems that you just have to write out and not do in your head... duh oh

Thank you very much!

Answer 16

when you have factor a hairy polynomial, you can use that insight that the constant term is always a product of the constant terms of the factors to make educated guesses as to what might be a factor

Answer 17

Yeah, there are a lot of problems like this where you just write down one thing and write down another thing like it and say "what do I have to do to make these match?"

Answer 18

Understood, thanks again!

Answer 19

you bet! I always like watching the "oh, I see!" moments :-)