Problem 42 - Precalculus : Building Concepts and Connections

Question

Find the remainder when \(x^{3}-3\) is divided by \(x+1.\)

Step-by-Step Solution

Answer

The remainder when \(x^{3}-3\) is divided by \(x+1\) is -4.

Step 1: Arrange the polynomial for division

Write out the polynomial \(x^{3}-3\) over \(x+1\) in long division or synthetic division format.

Step 2: Start the Division

In long division, we divide the first term of the numerator polynomial (the dividend) by the first term of the divisor polynomial. In this case, it's \(x^{3}\) ÷ \(x\), which gives us \(x^{2}\). Multiply the divisor polynomial by this result and write the answer underneath the dividend polynomial, aligning like terms.

Step 3: Subtract and bring down

Subtract the result you've just written down from the dividend polynomial. You'll get a new polynomial. Bring down the next term from the dividend polynomial (in this case, -3 because the dividend has missing terms and the subtraction result is 0), and repeat the process. The calculation stops when the degree of the divisor is higher than the degree of the remaining polynomial (or number).

Step 4: Inference

The final result is either a polynomial (with a lesser degree than the divisor) or a constant term that cannot be further divided by the divisor. That's our remainder.

Key Concepts

Synthetic Division

Synthetic division is a shorthand, or quicker form of polynomial long division. It's especially useful when you're dividing a polynomial by a linear factor of the form \(x - c\). Let's break down this process using the exercise given.

For the polynomial \(x^{3}-3\) divided by \(x+1\), we're essentially looking to divide by \(x - (-1)\). Synthetic division allows you to simplify this task by using only the coefficients and the constant term from the divisor, which in this case is -1. You initially draw a sort of 'L' shape, where you'll place the coefficients of the polynomial to be divided.

For our exercise, this means you write down 1 (for \(x^{3}\)), 0 (for \(x^{2}\) and \(x\), as they are 'invisible' with coefficients of 0), and -3. The number -1 is placed to the left outside of the 'L'. You then bring down the leading 1 and proceed to multiply it with -1 and put the result in the next column. Follow the steps of synthetic division, and you will arrive at the remainder efficiently.

You can think of synthetic division as a more compact version of long division that reduces the complexity of the process and is visually less cluttered. The final answer, or remainder, is easily seen at the bottom of your synthetic division set-up, making it quick and manageable to confirm whether you've done the process correctly.

Remainder Theorem

The Remainder Theorem is a fascinating piece of algebra that states when a polynomial \(f(x)\) is divided by a linear factor of the form \(x - c\), the remainder is the value of the polynomial \(f(c)\). In simpler terms, you plug the value opposite of \(c\) into the polynomial to find the remainder.

In the context of our exercise, to apply the Remainder Theorem to the polynomial \(x^{3}-3\) divided by \(x+1\), we would evaluate \(f(-1)\). This translates to \( (-1)^{3} - 3 = -1 - 3 = -4 \). This -4 is the remainder of the division, a quick and magical shortcut indeed!

The beauty of the Remainder Theorem lies in its simplicity and practicality, especially during examinations or in solving polynomials swiftly. Rather than performing long division or even synthetic division, the Remainder Theorem gets you to the remainder with minimal computation, saving time and reducing potential errors.

Polynomial Division

Polynomial division, whether it's long division or another method such as synthetic division, is a process used to divide a polynomial by another polynomial of equal or lower degree. Think of it as long division for numbers extended to polynomials. Important for understanding polynomial functions, it plays a critical role in tasks such as factorizing polynomials and finding polynomial roots.

In the exercise provided, we see a polynomial of the third degree being divided by a first-degree polynomial. The steps mimic numerical long division: divide the leading term, multiply and subtract, bring down the next term, and continue until you cannot divide any further. The last term or polynomial you're left with is the remainder.

Importance of Polynomial Division in Algebra

Polynomial division is not only a fundamental skill in algebra, it's essential for understanding more complex concepts like rational expressions, partial fractions, and integration of rational functions. Mastering this skill enhances problem-solving flexibility in calculus and beyond, and it provides a strong foundation for tackling a wide variety of mathematical challenges.