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Chapter 2: Problem 49

Suppose \(p\) is a polynomial and \(t\) is a number. Explain why there is a polynomial \(G\) such that $$ \frac{p(x)-p(t)}{x-t}=G(x) $$ for every number \(x \neq t\).

Short Answer

Expert verified
By using the factor theorem and synthetic division, we can break down \(p(x)-p(t)\) into \(G(x)*(x-t)+R\). Evaluating at x=t gives us a remainder of 0. Therefore, \(p(x)-p(t) = G(x)*(x-t)\), implying that \(\frac{p(x)-p(t)}{x-t}=G(x)\) for every number \(x \neq t\).

Step by step solution

01

Understanding Factor Theorem

Factor theorem states that if p(x) is a polynomial of degree n and if p(t) = 0, then (x - t) is a factor of p(x). In this case, we are not necessarily given that p(t) = 0, but we can still use the concept of synthetic division to break down p(x) - p(t).
02

Synthetic Division

Let's perform synthetic division on p(x) - p(t), using (x - t) as the divisor. Since p(x) is a polynomial of any degree, when we perform synthetic division on it, we will get another polynomial, and let's call it G(x). So after performing synthetic division on (p(x) - p(t)) with (x - t), we get the following equation: \(p(x) - p(t) = G(x) * (x - t) + R\) Here, R is the remainder we get after performing the synthetic division.
03

Evaluating the Function at x=t

Now, let's consider the case when x = t. Substituting x = t into the equation, we get: \(p(t) - p(t) = G(t) * (t - t) + R\) This simplifies to: \(0 = 0 + R\) Thus, R=0.
04

Arriving at the Given Equation

Now, let's plug this back into the equation obtained after synthetic division: \(p(x) - p(t) = G(x) * (x - t) + 0\) This simplifies to the given equation: \(\frac{p(x)-p(t)}{x-t}=G(x)\) Hence, there exists a polynomial G such that the given equation holds for all x 鈮� t.

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