Simplify (x^3+4x^2-5x-20)/(x-4)

The problem provided is a polynomial long division question. You are asked to divide the cubic polynomial $ x^3 + 4x^2 - 5x - 20 $ by the linear polynomial $ x - 4 $. The goal is to simplify the given expression to find the quotient and possibly a remainder, which would be expressed in terms of a polynomial of a lower degree than the original dividend (the cubic polynomial). In essence, the task is to perform the division in much the same way you would with numbers, but using the rules of algebra for polynomials.

$\frac{x^{3} + 4 x^{2} - 5 x - 20}{x - 4}$

Answer

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Solution:

Step 1: Group terms and factor

Divide the polynomial into two groups and factor each one.

Step 1.1: Group terms

Separate the polynomial into two groups: $\frac{(x^3 + 4x^2) - (5x + 20)}{x - 4}$

Step 1.2: Factor out the GCF from each group

Extract the GCF from each group: $\frac{x^2(x + 4) - 5(x + 4)}{x - 4}$

Step 2: Factor by grouping

Factor out the common binomial factor $(x + 4)$: $\frac{(x + 4)(x^2 - 5)}{x - 4}$

Knowledge Notes:

To simplify a rational expression, we can use the method of factoring by grouping. Here's a breakdown of the relevant knowledge points:

Factoring by Grouping: This technique involves rearranging the terms of a polynomial into groups that have a common factor. Once the terms are grouped, the common factor can be factored out, simplifying the expression.
Greatest Common Factor (GCF): The GCF of a set of terms is the largest expression that divides all the terms. Factoring out the GCF from each group helps in simplifying the polynomial.
Polynomial Division: When dividing polynomials, if the numerator can be factored such that one of the factors is the same as the denominator, those factors can cancel each other out, simplifying the expression.
Simplifying Rational Expressions: The goal is to reduce the expression to its simplest form by canceling out common factors in the numerator and denominator. This process is similar to simplifying fractions.
Latex Formatting: In the solution provided, Latex is used to clearly render mathematical expressions. For example, $x^2$ is written as $x^2$in Latex to display $ x^2 $.

By applying these concepts, we can simplify the given rational expression by factoring by grouping and then canceling out the common binomial factor.