Factor theorem polynomials: If x+6 is a factor of 3x^2+bx-48, then b=?

[sgray12]

Hello, I need some help trying to find the factor theorem with these polynomials. Please excuse some improper terms. I'm very new to this and taking a Pre-cal class.
I was presented with this equation:
If (x+6) is a factor of 3x^2+bx-48 then b=?

Thanks for your help

 

[Dr.Peterson]

Hello, I need some help trying to find the factor theorem with these polynomials. Please excuse some improper terms. I'm very new to this and taking a Pre-cal class.
I was presented with this equation:
If (x+6) is a factor of 3x^2+bx-48 then b=?

Thanks for your help

"The factor theorem" is a fact about factors; you need to use it, not find it.

What the theorem says (assuming it's the same theorem I have in mind) is that if (x-c) is a factor of a polynomial f, then f(c) = 0.

You want to find the value of the constant b if 3x^2+bx-48 = 0 when x=... what?

Take that hint, and see what you can do with it. Then show us whatever you were able to do.

 

[Steven G]

Dr Peterson gave you a great hint.
Here is a different way to think about it.
Since 3x^2+bx-48 is a quadratic, then it follows that if 3x^2+bx-48 factors it will factor to linear factors.

Since (x+6) IS a (linear) factor of 3x^2+bx-48, then 3x^2+bx-48 is factorable.

So 3x^2+bx-48 = (x+6)(rx + s). It should be clear that rx^2 = 3x^2 and 6s = -48.
Continue from here.

 

[The Highlander]

The Remainder Theorem states: When a polynomial f(x) is divided by (x – h) the remainder is f(h).

However, if f(h) = 0 then there is no remainder and (x – h) therefore divides evenly into the polynomial (ie: there is nothing left over) and so, in that case, (x – h) is, clearly, a factor of f(x).

Which leads us to…

The Factor Theorem which states: (x – h) is a factor of polynomial f(x) \(\displaystyle \Leftrightarrow\) f(h) = 0.

("\(\displaystyle \Leftrightarrow\)" means: "if and only if")​

But it is very important to note that it is (x minus h) that is a factor of f(x) when f(h) = 0!

Now, to make use of @Dr.Peterson's hint you simply need to decide what value for x you need to substitute into the equation and then determine what value "b" must take for the expression then to equal zero.

Please be sure to show us what answer you get for "b".