Problem 32 - Algebra 2

Question

Simplify. $$ \frac{x^{4}+x^{2}-3 x+5}{x^{2}+2} $$

Step-by-Step Solution

Answer

The simplified form is \( x^2 - 1 + \frac{-3x + 7}{x^2 + 2} \).

Step 1: Analyze the Polynomial

We begin by identifying the expression to be simplified: \( \frac{x^{4}+x^{2}-3x+5}{x^{2}+2} \). We can see that the numerator is a polynomial of degree 4 and the denominator is a polynomial of degree 2. Our goal is to simplify this expression, or to divide if possible.

Step 2: Setup Polynomial Long Division

To simplify the fraction, we perform polynomial long division of \( x^4 + x^2 - 3x + 5 \) by \( x^2 + 2 \). Write \( x^2 + 2 \) (the divisor) on the outside and \( x^4 + x^2 - 3x + 5 \) (the dividend) inside the division symbol.

Step 3: Divide the Leading Terms

Divide the leading term of the dividend, \( x^4 \), by the leading term of the divisor, \( x^2 \). The result is \( x^2 \). This is the first term of the quotient.

Step 4: Multiply and Subtract

Multiply the entire divisor \( x^2 + 2 \) by \( x^2 \) (the term from the previous step), resulting in \( x^4 + 2x^2 \). Subtract this product from the original dividend, giving \( (x^4 + x^2 - 3x + 5) - (x^4 + 2x^2) = -x^2 - 3x + 5 \).

Step 5: Repeat Division

The new dividend is \( -x^2 - 3x + 5 \). Divide the leading term, \( -x^2 \), by \( x^2 \) to get \( -1 \). This becomes the next term in the quotient.

Step 6: Multiply and Subtract Again

Multiply \( x^2 + 2 \) by \( -1 \) to get \( -x^2 - 2 \). Subtract this from \( -x^2 - 3x + 5 \) to get \( (-x^2 - 3x + 5) - (-x^2 - 2) = -3x + 7 \).

Step 7: Write Final Quotient and Remainder

The quotient is \( x^2 - 1 \) and the remainder is \( -3x + 7 \). The simplified form of the original expression is: \[ x^2 - 1 + \frac{-3x + 7}{x^2 + 2} \].

Key Concepts

Simplifying Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying these expressions often involves polynomial long division, especially when you have a complex polynomial in the numerator and a simpler one in the denominator.

In our example, we started with the expression \( \frac{x^4+x^2-3x+5}{x^2+2} \). The key to simplifying this expression lies in dividing the numerator by the denominator, similar to regular numerical division but with polynomials.

Steps to simplify include:

• Identify the degree of both polynomials.
• Use polynomial long division to divide the polynomials.
• Simplify the expression with any quotient and incorporate the remainder properly.

Degree of Polynomial

The degree of a polynomial is a crucial concept when performing polynomial long division. It is the highest power of the variable in the polynomial.

In the expression \( x^4 + x^2 - 3x + 5 \), the degree is 4 because the highest exponent in the polynomial is 4. Similarly, for \( x^2 + 2 \), the degree is 2. Understanding these degrees helps determine how the division will proceed.

When dividing, compare the leading terms (the terms with the highest degree). This aids in finding how many times the divisor can "fit" into segments of the dividend, helping you form the quotient.

Quotient and Remainder

After performing polynomial long division, you end up with a quotient and sometimes a remainder. These are significant results in the division process because they show how the polynomial division simplifies the original rational expression.

In this exercise, dividing \( x^4 + x^2 - 3x + 5 \) by \( x^2 + 2 \) gave us a quotient of \( x^2 - 1 \) and a remainder of \( -3x + 7 \).

The quotient \( x^2 - 1 \) shows the main simplified term of the expression, while the remainder \( \frac{-3x + 7}{x^2 + 2} \) represents the leftover part that couldn’t be evenly divided. All together, they form the simplified expression: \[ x^2 - 1 + \frac{-3x + 7}{x^2 + 2} \] This simplification can make further algebraic manipulators easier to handle and understand.