Problem 53 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \( x^3 + 9x^2 + 41x + 210 + \frac{1050}{x-5} \).

Step 1: Identify the Division

Notice that the expression given is a fraction. The task is to simplify this fraction: \( \frac{x^{4} + 4x^{3} - 4x^{2} + 5x}{x - 5} \). The numerator is a polynomial of degree 4, and the denominator is a polynomial of degree 1.

Step 2: Perform Polynomial Long Division

We will divide \( x^{4} + 4x^{3} - 4x^{2} + 5x \) by \( x - 5 \) using polynomial long division. Start by dividing the highest degree term of the numerator, \( x^4 \), by the highest degree term of the denominator, \( x \). This gives \( x^3 \). Multiply \( x^3 \) by \( x - 5 \), resulting in \( x^4 - 5x^3 \). Subtract this from the original polynomial to get \( 9x^3 - 4x^2 + 5x \).

Step 3: Continue the Division Process

Next, divide \( 9x^3 \) by \( x \), which gives \( 9x^2 \). Multiply \( 9x^2 \) by \( x - 5 \) to get \( 9x^3 - 45x^2 \). Subtract this from the current remainder \( 9x^3 - 4x^2 + 5x \), resulting in \( 41x^2 + 5x \).

Step 4: Divide Each Term Until Remainder is Relatively Lower Than Denominator

Next, divide \( 41x^2 \) by \( x \) to get \( 41x \). Multiply \( 41x \) by \( x - 5 \) to produce \( 41x^2 - 205x \). Subtract this result from \( 41x^2 + 5x \), yielding \( 210x \). Divide \( 210x \) by \( x \), giving \( 210 \). Multiply \( 210 \) by \( x - 5 \) to get \( 210x - 1050 \). Subtract this from \( 210x \), resulting in \( 1050 \).

Step 5: Determine Quotient and Remainder

The quotient of the division process is \( x^3 + 9x^2 + 41x + 210 \), and the remainder is \( 1050 \). Therefore, the simplified fraction is:\[\frac{x^{4}+4 x^{3}-4 x^{2}+5 x}{x-5} = x^3 + 9x^2 + 41x + 210 + \frac{1050}{x-5}\]

Step 6: Simplified Expression

The expression has been simplified as a polynomial plus a remainder over the original denominator. Since \( \frac{1050}{x-5} \) cannot be simplified further, this is the fullest decomposition of the expression.

Key Concepts

Polynomial Simplification

Polynomial simplification is the process of reducing a complex polynomial expression into its simplest form. This involves expression manipulation, such as combining like terms, canceling common factors, and using special algebraic identities. The goal is to make the polynomial shorter and easier to understand.

• Start by identifying the polynomial's terms and their coefficients.
• Simplify by performing arithmetic operations (addition, subtraction) among like terms to combine them.
• Check for any common factors or special patterns, like the difference of squares, that can be factored out for further simplification.

For the problem at hand, polynomial simplification involves breaking down the fraction into a simpler polynomial plus a remainder.
This is achieved by reducing the original polynomial through division by another polynomial.

Division of Polynomials

The division of polynomials, especially with polynomial long division, is a method to divide one polynomial by another polynomial, yielding a quotient and a remainder. It's akin to traditional long division used with numbers, but involves monomials (terms) instead.

• Identify the highest degree terms from both the numerator and denominator polynomials.
• Divide the leading coefficient of the numerator by the leading coefficient of the denominator.
• Multiply the entire denominator by the result obtained and subtract it from the original numerator.
• Repeat this process with the new polynomial formed until the degree of the remainder is less than that of the denominator.

In our example, we divided each degree level of the polynomial until we received a remainder.
The quotient was found through repeated division of the highest degree terms.

Remainder Theorem

The Remainder Theorem is a valuable tool in algebra that allows us to find the remainder of a polynomial division without performing the complete division. It states that if a polynomial \( f(x) \) is divided by a linear divisor \( x-a \), the remainder of this division is simply \( f(a) \).

• This theorem can be used to check the result of polynomial long division quickly.
• It helps in determining if a particular value makes the polynomial equal to zero, indicating a root.

In our division example, knowing the remainder is important because it shows what remains when the polynomial is divided by \( x - 5 \).
Thus, even if not fully factored, the polynomial expression is simplified in its most understandable form.