MHB ACT Problem: Determine Constant In Polynomial Given Factor

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To determine the constant z in the polynomial 2x^2 - 4x - z, given that (x-4) is a factor, we set f(4) = 0. Substituting x = 4 into the polynomial yields the equation 32 - 16 - z = 0. Simplifying this leads to 16 - z = 0, which results in z = 16. Therefore, the value of z is 16. This method effectively uses the factor theorem to solve for the constant.
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Assume that (x-4) is a factor of 2x^2-4x-z. What is the value of z?

How would you set it up to use the foil method?
 
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Re: ACT problem

Let:

$$f(x)=2x^2-4x-z$$

Now, if $x-4$ is a factor of $f$, then we must have:

$$f(4)=0$$

So, this allows us to write:

$$2(4)^2-4(4)-z=0$$

Can you proceed?
 
Re: ACT problem

MarkFL said:
Let:

$$f(x)=2x^2-4x-z$$

Now, if $x-4$ is a factor of $f$, then we must have:

$$f(4)=0$$

So, this allows us to write:

$$2(4)^2-4(4)-z=0$$

Can you proceed?

32-16-z+=0; 16-z=0; 16=z
 
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