Question

suppose that F,G and Q are polynomials and \(\frac{{F(x)}}{{Q(x)}} = \frac{{G(x)}}{{Q(x)}}\)for all x except when\(Q(x) = 0\).prove that\(F(x) = G(x)\)for all x.

Short answer

we can prove that \(F(x) = G(x)\) for all x by using cross multiplication

Step-by-step solution

Write the given data

Let us consider

\(\frac{{F(x)}}{{Q(x)}} = \frac{{G(x)}}{{Q(x)}}\)

We know that all the functions are polynomials.

By the definition we know that F,G, Q are all continuous

Apply the cross multiplication

So therefore we can cross multiply equations.

\(F(x)Q(x) = G(x)Q(x)\)and\(Q(x) \ne 0\)

Since all are continuous for all x, so we can divide by\(Q(x)\)to obtain

\(F(x) = G(x)\)

Hence, we can prove that \(F(x) = G(x)\) for all x.