Problem 53 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient and remainder. $$ \frac{x^{4}-22}{x+2} $$

Step-by-Step Solution

Answer

Quotient: \(x^3 - 2x^2 + 4x - 8\), Remainder: \(-6\).

Step 1: Set Up Synthetic Division

Write down the coefficients of the polynomial \(x^4 - 22\). Since there are no terms for \(x^3\), \(x^2\), and \(x^1\), their coefficients are 0. So, we use the coefficients: \([1, 0, 0, 0, -22]\). The divisor is \(x + 2\), so we use \(-2\) in synthetic division.

Step 2: Perform Synthetic Division

1. Bring down the first coefficient 1.2. Multiply \(1\) by \(-2\) and write the result under the next coefficient.3. Add this to the coefficient above and repeat for each coefficient. - Multiply: \(1 \times -2 = -2\) - Add: \(0 + (-2) = -2\) - Multiply: \(-2 \times -2 = 4\) - Add: \(0 + 4 = 4\) - Multiply: \(4 \times -2 = -8\) - Add: \(0 + (-8) = -8\) - Multiply: \(-8 \times -2 = 16\) - Add: \(-22 + 16 = -6\)

Step 3: Identify the Quotient and Remainder

From synthetic division, you get the sequence \([1, -2, 4, -8 | -6]\). These numbers represent the coefficients of the quotient and the remainder, respectively:- Quotient: \(1x^3 - 2x^2 + 4x - 8\)- Remainder: \(-6\)

Step 4: Write the Final Result

The result of the division is written as:\[\frac{x^4 - 22}{x + 2} = x^3 - 2x^2 + 4x - 8 - \frac{6}{x+2}\]

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials and is similar to long division with numbers. When you divide a polynomial by another polynomial, the idea is to find a quotient and a remainder, just like in numerical division.
This division simplifies the expression and helps in understanding the behavior of polynomials. There are mainly two types of polynomial division: traditional long division and synthetic division.
Synthetic division is a simplified form of polynomial division that is particularly useful when dividing by linear factors of the form \(x - c\). It's faster and involves fewer steps compared to long division.

• Advantage: Faster and less tedious.
• Disadvantage: Only works for divisors of the form \(x - c\).

Understanding polynomial division is crucial for solving higher-degree polynomial equations, factoring polynomials, and simplifying expressions.

Remainder Theorem

The remainder theorem provides a quick way to find the remainder of a polynomial division without performing the entire division process. It states that if a polynomial \(f(x)\) is divided by a linear divisor of the form \(x - c\), the remainder is simply \(f(c)\).
This theorem is remarkably useful because it allows us to evaluate polynomials and find remainders by substitution.

• To apply: Substitute the value \(c\) into the polynomial \(f(x)\).
• The value of the polynomial at \(x = c\) gives the remainder.

In the case of the given exercise, after performing synthetic division, we found that the remainder is \(-6\). This should match the value you'd obtain by substituting \(-2\) in the polynomial \(x^4 - 22\). Understanding the remainder theorem allows students to verify their division results and improve their grasp of polynomial functions.

Algebraic Expressions

Algebraic expressions consist of terms that include numbers, variables, and operations. They form the building blocks of algebra, representing values or expressions in mathematical problems.
In the case of the polynomial division \(\frac{x^4 - 22}{x+2}\), the polynomial \(x^4 - 22\) is an algebraic expression.
It's important to understand how to manipulate these expressions, whether by addition, subtraction, multiplication, or division.

• Consist of terms: Can be constants, variables, or coefficients.
• Operations: How terms are manipulated.

Mastering algebraic expressions allows students to tackle complex problems involving equations, simplify expressions, and carry out polynomial operations. Moreover, grasping this concept is foundational for learning higher mathematics, where algebra is applied in calculus, linear algebra, and beyond.