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Let f, g, and h be polynomials such that h(x) = f(x) * g(x). If the constant term of f(x) is -4 and the constant term of h(x) is 3, what is g(0)?

 May 26, 2020
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f,g, and h are all polynomial functions, so they all adhere to the following general form shown below.

 

\(f(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_2x^2+a_1x^1+a_0\\ g(x)=b_nx^n+b_{n-1}x^{n-1}+b_{n-2}x^{n-2}+...+b_2x^2+b_1x^1+b_0\\ h(x)=c_nx^n+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+...+c_2x^2+c_1x^1+c_0 \)

 

The constant term of \(f(x)\), represented as \(a_0\) in the general form, is \(-4\). In the function \(h(x)\), the constant term is 3. One way to guarantee that a polynomial function will output its constant term is to substitute in \(x=0\) for that polynomial. 

 

\(h(0)=f(0)*g(0)\\ c_0=a_0*b_0\\ 3=-4b_0\\ b_0=\frac{-3}{4}\\ \therefore g(0)=b_0=\frac{-3}{4}\)

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 May 26, 2020

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