Problem 32 - Algebra and Trigonometry

Question

25-38 . Find the quotient and remainder using synthetic division. $$ \frac{x^{4}-x^{3}+x^{2}-x+2}{x-2} $$

Step-by-Step Solution

Answer

The quotient is \(x^3 + x^2 + 3x + 5\) with a remainder of 12.

Step 1: Set up Synthetic Division

The divisor is \(x - 2\), hence the root is \(2\). Write \(2\) to the left outside the synthetic division bracket. Write the coefficients of the dividend \(x^4 - x^3 + x^2 - x + 2\) inside the bracket, which are \(1, -1, 1, -1, 2\).

Step 2: Bring Down the First Coefficient

Bring the first coefficient \(1\) straight down below the horizontal line.

Step 3: Perform Synthetic Multiplication and Addition

Multiply the root \(2\) by the value just written below the horizontal line \((1)\). Write the result \(2\) under the next coefficient \((-1)\). Add \(-1 + 2 = 1\) and write the result below the line.

Step 4: Continue Synthetic Multiplication and Addition

Multiply the root \(2\) by the new result \((1)\). Write \(2\) under the next coefficient \((1)\), add these to get \(3\). Continue by multiplying and adding like this: \(2 \cdot 3 = 6\) leading to \(-1 + 6 = 5\), \(2 \cdot 5 = 10\) leading to \(2 + 10 = 12\).

Step 5: Identify Quotient and Remainder

The numbers below the line, apart from the last one, represent the coefficients of the quotient polynomial. Since we started with an \(x^4\) polynomial, the quotient is \(x^3 + x^2 + 3x + 5\). The last number, \(12\), is the remainder.

Key Concepts

Polynomial Division

Polynomial division is a process similar to long division but applied to polynomials. It's a method used to divide a polynomial (the dividend) by another polynomial (the divisor). When using polynomial division, our goal is to simplify the expression or find the quotient and remainder. For example, dividing the polynomial \(x^4 - x^3 + x^2 - x + 2\) by \(x - 2\) requires calculating how many times \(x - 2\) can "fit" into the higher-degree polynomial.

This can be a complex process, especially with higher-degree polynomials, but methods like synthetic division simplify the calculations.
Synthetic division is a shortcut which is specifically designed for dividing polynomials by divisors of the form \(x - c\). This approach focuses on the coefficients, making it much faster and easier than the traditional method.

Quotient and Remainder

In any division process, whether dealing with numbers or polynomials, the result can be expressed as a quotient and a remainder. The quotient is what you get when you divide the dividend by the divisor without considering the remainder. The remainder, on the other hand, is what's left over after completing the division.

• The quotient is expressed as a polynomial if the division involves polynomials, having one less degree than the dividend.
• The remainder is a lower-degree polynomial than the divisor, often a constant when dividing by \(x - c\).

For instance, when we perform synthetic division on \(x^4 - x^3 + x^2 - x + 2\) by \(x - 2\), the quotient is \(x^3 + x^2 + 3x + 5\) and the remainder is \(12\). This means the division can be written as: \[ (x^4 - x^3 + x^2 - x + 2) = (x - 2)(x^3 + x^2 + 3x + 5) + 12 \] This demonstrates that even as polynomials, the division has been balanced.

Roots of Polynomials

The roots of a polynomial are the values for which the polynomial evaluates to zero. They are particularly essential when performing polynomial division as they might help us verify if we've correctly divided the polynomial.

Consider the divisor \(x - c\) in synthetic and polynomial division. The root is \(c\). If \(x - c\) divides a polynomial with zero remainder, then \(c\) is a root of that polynomial. However, if there is a non-zero remainder, as seen in our example, it tells us that \(c\) is not a root. Here, when dividing \(x^4 - x^3 + x^2 - x + 2\) by \(x - 2\), because the remainder is 12, we see that 2 is not a root of the polynomial.

Working with roots and understanding their role can help in long-term algebraic manipulation and solving polynomial equations.

Algebraic Expressions

An understanding of algebraic expressions is fundamental when tackling polynomial division and other algebraic tasks. Algebraic expressions are combinations of variables, numbers, and operators (like addition and subtraction).

They represent mathematical relationships and can be used to model real-life problems. When dividing, multiplying, or simplifying algebraic expressions, we break them down into simpler forms.

For example:

• In our division, \(x^4 - x^3 + x^2 - x + 2\) is the algebraic expression or polynomial being divided.
• Synthetic division further simplifies working with this polynomial since it reduces the expression evaluation to simple arithmetic operations.

Understanding the structure and operations on algebraic expressions aids in simplifying complex polynomials and solving intricate algebraic problems.