Question

What is the remainder when \(f(x)=2 x^{20}-8 x^{10}+x-2\) is divided by \(x-1 ?\)

Short answer

The remainder is -7.

Step-by-step solution

Use the Remainder Theorem

The Remainder Theorem states that the remainder when a polynomial, say \(f(x)\), is divided by \(x-a\) is \(f(a)\). In this case, we need to find the remainder when \(f(x) = 2x^{20} - 8x^{10} + x - 2\) is divided by \(x-1\). Here, \(a=1\).

Evaluate the Polynomial

Substitute \(x=1\) into the polynomial \(f(x) = 2x^{20} - 8x^{10} + x - 2\). That is, compute \(f(1)\).

Perform the Calculation

Substitute \(x=1\) into the expression: \[ f(1) = 2(1)^{20} - 8(1)^{10} + 1 - 2 \]Simplify the terms:\[ f(1) = 2(1) - 8(1) + 1 - 2 = 2 - 8 + 1 - 2 = -7 \]Therefore, the remainder is -7.

Key concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. It's similar to long division with numbers, except it involves variables and their exponents. The process can help in breaking down complex polynomials into simpler ones and finding remainders.

When we talk about dividing a polynomial by a linear divisor (like \(x - a\)), the Remainder Theorem offers a quick shortcut to find the remainder. This is extremely useful since performing long division manually can be cumbersome, especially with high-degree polynomials.

Remainder Calculation

Calculating the remainder of polynomial division can be simplified using the Remainder Theorem. This theorem states that if you divide a polynomial \(f(x)\) by \(x-a\), the remainder is simply \(f(a)\). This saves a lot of time and effort.

For instance, given the polynomial \(f(x) = 2x^{20} - 8x^{10} + x - 2\), and asked to find the remainder when it’s divided by \(x-1\), the theorem tells us to evaluate \(f(1)\). By substituting \(1\) for \(x\) in the polynomial, we quickly find the remainder:
\[ f(1) = 2(1)^{20} - 8(1)^{10} + 1 - 2 = 2 - 8 + 1 - 2 = -7 \].
This remainder is much easier to find than using long division.

Evaluating Polynomials

Evaluating polynomials refers to the process of finding the value of the polynomial at a specific point. It involves substituting the value for \(x\) and simplifying based on the polynomial's terms.

For example, to evaluate \(f(x) = 2x^{20} - 8x^{10} + x - 2\) at \(x=1\), we substitute \(1\) in place of \(x\) and perform the arithmetic:
\[f(1) = 2(1)^{20} - 8(1)^{10} + 1 - 2 = 2 - 8 + 1 - 2 = -7\]
This evaluation helps in applications like calculating remainders using the Remainder Theorem and finding specific values which could be needed in graphing or solving polynomial equations.