Problem 61 - College Algebra

Question

State the Remainder Theorem.

Step-by-Step Solution

Answer

The Remainder Theorem states that when a polynomial f(x) is divided by \(x-a\), the remainder is \(f(a)\). It provides a shortcut to finding the remainder of a polynomial division by substitution.

Step 1: State the Remainder Theorem

The Remainder Theorem states that if a polynomial f(x) is divided by \(x-a\), then the remainder is \(f(a)\). This theorem is very useful when the divisor is a linear polynomial.

Step 2: Explain the Basic Principle

In practice, to use the Remainder Theorem, substitute the value of \(a\) from the divisor \(x-a\) into the polynomial \(f(x)\). The resulting value is the remainder when \(f(x)\) is divided by \(x-a\). This concept is much simpler and quicker than performing polynomial long division.

Step 3: Provide an Example

To illustrate this, consider \(f(x)=2x^3-3x^2+x+1\) and we are dividing it by \(x-1\). Substituting \(x=1\) into \(f(x)\) gives \(2*1^3-3*1^2+1+1 = 1\), so the remainder when \(f(x)\) is divided by \(x-1\) is 1.