Simplify (z^2+zx+5z+5x)/(z+5)
The problem presented is a simplification problem involving polynomial long division or algebraic fraction reduction. You are tasked with simplifying the given algebraic expression by dividing the polynomial in the numerator, z^2 + zx + 5z + 5x, by the binomial in the denominator, z + 5. Essentially, you are asked to find a simpler equivalent expression, which may potentially be a polynomial or another algebraic expression, that does not have the denominator z + 5 when it is fully simplified.
$\frac{z^{2} + z x + 5 z + 5 x}{z + 5}$
Answer
Solution:
Step 1: Simplify the numerator
Step 1.1: Extract the greatest common factor from each term grouping
Step 1.1.1: Group similar terms together
$\frac{(z^2 + zx) + (5z + 5x)}{z + 5}$
Step 1.1.2: Identify and factor out the GCF from each group
$\frac{z(z + x) + 5(z + x)}{z + 5}$
Step 1.2: Factor by grouping
$\frac{(z + x)(z + 5)}{z + 5}$
Step 2: Eliminate the common factor
Step 2.1: Remove the common factor $(z + 5)$
$\frac{(z + x)\cancel{(z + 5)}}{\cancel{z + 5}}$
Step 2.2: Simplify the expression
$z + x$
Knowledge Notes:
The problem-solving process involves simplifying a rational expression by factoring and canceling common factors. Here are the relevant knowledge points:
Factoring: This is the process of breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. In this case, factoring involves grouping terms that have a common factor and then extracting that factor.
Greatest Common Factor (GCF): The GCF of a set of terms is the largest expression that divides all of them without leaving a remainder. In the context of polynomials, it's the highest degree of common variables with the smallest coefficient that can be factored out.
Factoring by Grouping: This technique is used when a polynomial doesn鈥檛 have a common factor in all terms but can be grouped in a way that allows us to factor by pairs or groups. In this problem, the terms are grouped into two pairs, each of which shares a common factor.
Canceling Common Factors: In a fraction, if the numerator and denominator share a common factor, it can be canceled out. This is based on the property that a fraction remains unchanged if both the numerator and denominator are divided by the same non-zero number.
Simplifying Rational Expressions: The process of simplifying rational expressions involves factoring both the numerator and the denominator and then canceling out common factors. The goal is to write the expression in its simplest form.
LaTeX Formatting: To clearly present mathematical expressions, LaTeX is used. It is a typesetting system that is widely used for formatting complex mathematical formulas and equations.
By applying these concepts, the given rational expression is simplified by factoring the numerator, identifying the common factor with the denominator, and then canceling this common factor to reach the simplest form of the expression.
