Problem 25 - College Algebra with Modeling and Visualization

Question

Divide the expression. $$\frac{5 x^{4}-2 x^{2}+6}{x^{2}+2}$$

Step-by-Step Solution

Answer

The result is \( 5x^2 - 12 + \frac{30}{x^2 + 2} \).

Step 1: Set Up Division

To divide the polynomial \( \frac{5x^4 - 2x^2 + 6}{x^2 + 2} \), we'll use polynomial long division. Set up the division by writing \( 5x^4 - 2x^2 + 6 \) as the dividend and \( x^2 + 2 \) as the divisor.

Step 2: Divide Leading Terms

Divide the leading term of the dividend \( 5x^4 \) by the leading term of the divisor \( x^2 \) to get the first term of the quotient: \( 5x^2 \). This is obtained by \( \frac{5x^4}{x^2} = 5x^2 \).

Step 3: Multiply and Subtract

Multiply \( 5x^2 \) by the entire divisor \( x^2 + 2 \) to get \( 5x^4 + 10x^2 \). Subtract this from the original dividend: \((5x^4 - 2x^2 + 6) - (5x^4 + 10x^2) = -12x^2 + 6\).

Step 4: Repeat Division

Now repeat the process for \( -12x^2 \). Divide \( -12x^2 \) by \( x^2 \) to get \( -12 \). This is added to the quotient.

Step 5: Multiply and Subtract Again

Multiply \( -12 \) by \( x^2 + 2 \) to get \( -12x^2 - 24 \). Subtract from \( -12x^2 + 6 \): \((-12x^2 + 6) - (-12x^2 - 24) = 30\).

Step 6: Conclusion on Remainder

The remainder after the division process is \( 30 \) and cannot be further divided by \( x^2 + 2 \). So, the final result of the division is \( 5x^2 - 12 + \frac{30}{x^2 + 2} \).

Key Concepts

Polynomial Division

Polynomial division is like long division with numbers, but instead we work with algebraic expressions. It's used to divide two polynomials, called the dividend and the divisor, to find the quotient and sometimes a remainder. The goal is to simplify a complex polynomial into a more manageable form.

Here's how it works:

• Set up the division with the larger polynomial as the dividend and the smaller one as the divisor.
• Divide the first term of the dividend by the first term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this term, then subtract the result from the dividend.
• Repeat the process with the new polynomial (what's left after subtraction) until the remainder is smaller than the divisor or zero.

This process helps break down complex polynomial expressions, making it easier to visualize and handle within algebraic equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a specific value or values. They're fundamental in algebra, allowing us to generalize mathematical problems using symbols.

Characteristics of algebraic expressions include:

• A combination of constants (numbers) and variables (like \( x \) and \( y \)).
• Involve operations such as addition, subtraction, multiplication, and division.
• Can include exponents or powers like \( x^2 \).

Expressions like \( 5x^4 - 2x^2 + 6 \) show how our variables (e.g., \( x \)) interact with multipliers and constants. Understanding these expressions is crucial since they form the basis for solving equations and inequalities in algebra. They provide a concise way to represent mathematical ideas or rules.

Remainders in Division

In polynomial division, just as with integer division, a remainder might be left over. The remainder is what is "left" when nothing more can be divided by the divisor. Understanding remainders is important because they show how well a polynomial divides into another.

Key points about remainders in polynomial division:

• The remainder has a degree lower than that of the divisor.
• If the remainder is zero, the divisor perfectly divides the dividend.
• If not, the remainder is included in the final answer as a fraction, such as \( \frac{remainder}{divisor} \).

In our example, after performing the long division, we find a remainder of 30, resulting in the expression \( 5x^2 - 12 + \frac{30}{x^2 + 2} \). This tells us the original polynomial is almost perfectly divisible by \( x^2 + 2 \) but leaves behind a small part—our remainder.