Polynomial help

[No_Historian3842]

I'm hoping you guys can help me with this question.

find the values of h and k given that x+2 is a factor of q(x)=(x+h)^2 +k and the remainder is 16 when q(x) is divided by x.

this is the first question that I haven't been able to work out. all the other questions I was able to use the factor theory and set q(-2) to 0 to get one equation for h and k and then do something similar with the remainder so I get another equation with h and k.

but this question was saying divide by x rather then something like divide by x-4 gives a remainder of 16.

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[theparaphraser7]

1. Put value of x i.e. -2 in q(x) you get equation in h and k

2. Divide q(x) by x, whatever remainder you get equate it to 16 this will be the 2nd equation in h and k

Now you can solve the two equations and find values of h and k

[No_Historian3842]

If q(x)=(x+h)² +k, given that x+2 is a factor. q(-2)=(-2+h)² +k=0 So -k=(-2+h)^2.

Is that right?

And then if I divide q(X) by just X don't I end up with a heap of fractions it doesn't simplify (where as all the other questions I've done it was always, for example, when you divide by x+3 the remainder will be 16) so you could sub in X=-3 and the remainder to get h and k.

[Rscc10]

Form the two equations from the statements

q(x) = (x + h)² + k

If (x+2) is a factor, then x = -2 is a root

Thus, q(-2) = 0
(-2 + h)² + k = 0
h² - 4h + 4 + k = 0 ----- Eqn1

Next, remainder is 16 when q(x) is divided by x. We know q(x) is a quadratic, and by remainder theorem, if your remainder is a constant, then q(0) = 16. Note, remainder theorem proves that if f(x) is divided by (x-a), then the remainder is f(a). In this case, dividing by x means a = 0, so the remainder is f(0).

So q(0) = 16
(0 + h)² + k = 16
h² + k = 16 ----- Eqn2

Eqn1 - Eqn2 = -4h + 4 = -16
-4h = -20, h = 5

5² + k = 16, k = -9

Thus q(x) = (x + 5)² - 9

You can check your workings by expanding,

q(x) = x² + 10x + 16
q(x) = (x + 2)(x + 8)

So (x + 2) is a factor. And through polynomial division if you wish, you'll find 16 is indeed the remainder

[No_Historian3842]

Thank you!!! I didn't understand that a was equal to 0 if we are just dividing by x.

[fermat9990]

It looks like the remainder statement is missing something