Problem 5 - Precalculus : Building Concepts and Connections

Question

Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$x^{3}-3 x^{2}+2 x-4 ; x+2$$

Step-by-Step Solution

Answer

The quotient of \(x^{3}-3 x^{2}+2 x-4\) divided by \(x+2\) is \(x^2 - 5x + 12\) and the remainder is -20.

Step 1: Set Up the Synthetic Division

The first step is to set up the Synthetic Division. Write down the coefficients of the polynomial being divided, which are 1 (the coefficient of \(x^{3}\)), -3 (the coefficient of \(x^{2}\)), 2 (the coefficient of \(x\)) and -4. Also write down the number you'll use for the division, which is -2 (the opposite value of the constant term from \(x+2)\)

Step 2: Carry out the Synthetic Division

Now, begin the Synthetic Division process. Bring the first number (1) straight down. Then, multiply that number by -2 and write the result under the next number. Add the column and write the result below, in the next row. Continue this process until you have dealt with all the members of the row.

Step 3: Interpret the Result

The final row of numbers represents the coefficients of the quotient polynomial. The last number is the remainder. So we have a quotient of \(x^2 - 5x + 12\) and a remainder of -20.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. It is similar to long division but is done with algebra instead of numbers. There are various ways to perform polynomial division, and one of the most common is synthetic division, especially when dividing polynomials by a linear binomial.When dealing with polynomial division, our main goal is to find two new expressions — the quotient and the remainder. These results help express the division in a simplified form. The expression can be rearranged as:Given a dividend polynomial \( P(x) \) and a divisor \( D(x) \), the division can be described by the equation:\[P(x) = Q(x) \cdot D(x) + R(x)\]Where:

• \( Q(x) \) is the quotient
• \( R(x) \) is the remainder

This equation is quite similar to elementary school math where a number can be expressed in terms of its quotient and remainder when divided by another number.

Quotient and Remainder

In polynomial division, just as in regular division, the quotient and remainder are the results we obtain. Let's break them down further:

• Quotient: The quotient in polynomial division is the polynomial obtained after dividing the original polynomial (dividend) by the divisor. It is typically one degree less than the degree of the dividend when using synthetic division. For example, dividing the polynomial \(x^3 - 3x^2 + 2x - 4\) by \(x + 2\) yields the quotient \(x^2 - 5x + 12\).
• Remainder: The remainder is the polynomial that remains once the division is complete. In some cases, the remainder may be zero, indicating that the dividend is evenly divisible by the divisor. In this particular example, the remainder is -20, which means after division, -20 is left over when \(x^3 - 3x^2 + 2x - 4 \) is divided by \(x + 2\).

The understanding of quotient and remainder is critical in algebra since it helps to solve equations and simplify expressions. The quotient can usually be used in further calculations, while the remainder is added to while resuming calculations after division.

Polynomial Coefficients

Polynomial coefficients are the numbers in front of the terms in a polynomial. They play a crucial role in polynomial division since they define the terms being divided. For instance, in the polynomial \(x^3 - 3x^2 + 2x - 4\), the coefficients are:

• 1 for \(x^3\)
• -3 for \(x^2\)
• 2 for \(x\)
• -4 for the constant term

Knowing these coefficients is essential for setting up synthetic division. Synthetic division simplifies division by using these coefficients instead of rewriting the entire polynomial repeatedly.When setting up synthetic division, you list the coefficients in order, and use the constant term from the divisor, \(x + 2\). You take the opposite sign of the constant term (which is -2 in our example) to perform the division process.Being familiar with polynomial coefficients is fundamental in algebra as they determine how each term contributes to the polynomial's behavior and are the heart of many algebraic processes.