Question

When a polynomial function \(f\) is divided by \(x-c,\) the remainder is _______ .

Short answer

The remainder is \(f(c)\).

Step-by-step solution

- Understand the Problem

Determine what is being asked. The exercise gives a polynomial function \(f\) and involves dividing it by \(x - c\). The goal is to find the remainder of this division.

- Use the Remainder Theorem

The Remainder Theorem states that when a polynomial \(f(x)\) is divided by \(x - c\), the remainder of this division is equal to \(f(c)\).

- Write the Conclusion

According to the Remainder Theorem, if the polynomial function \(f\) is divided by \(x - c\), then the remainder is \(f(c)\). Therefore, the remainder is \(f(c)\).

Key concepts

Polynomial Division

Polynomial division is a process similar to long division with numbers, but it involves dividing one polynomial by another. The goal is to express the original polynomial as the product of the divisor polynomial and a quotient polynomial, plus a remainder. Here’s how it typically works:

• You start by determining the leading term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
• Then, you multiply the divisor by this leading term of the quotient and subtract this product from the dividend.
• Repeat these steps with the new polynomial formed after subtraction until the degree of the remaining polynomial is less than the degree of the divisor.

This final polynomial that you get after subtraction is called the

Remainder

In polynomial division, the remainder is what’s left over after completing the division process. When you divide polynomials, the remainder will always have a degree less than the divisor polynomial. If you are dividing by a polynomial of degree 1, like \(x - c\), the remainder is a constant term.
The Remainder Theorem makes this process simpler by providing a quick way to find the remainder. According to this theorem, when a polynomial \(f(x)\) is divided by \(x - c\), the remainder is simply the value of the polynomial evaluated at \(c\). This means:
\[ \text{Remainder} = f(c) \]This handy theorem allows you to bypass the entire division process and directly find the remainder by evaluating the polynomial at the given value.

Evaluation of Polynomial

Evaluating a polynomial at a certain value involves substituting that value into the polynomial and computing the result. For example, if you have a polynomial \(f(x) = 3x^2 + 2x + 1\) and you want to evaluate it at \(x = 2\), you substitute 2 into every instance of x:
\[ f(2) = 3(2)^2 + 2(2) + 1 \]Then, carry out the arithmetic:

• First, compute \(3(2)^2 = 3*4 = 12\),
• Then, compute \(2(2) = 4\)
• Finally, add these results and the constant term: \(12 + 4 + 1 = 17\)

This result, 17, is the value of the polynomial at \(x = 2\).The Remainder Theorem uses this concept directly to find the remainder of a polynomial divided by \(x - c\). Instead of performing polynomial division, just evaluate \(f(c)\) and that value is your remainder.