Problem 69 - Algebra 2

Question

Simplify each rational expression. $$ \frac{3 z^{4}+36 z^{3}+60 z^{2}}{3 z^{3}-3 z^{2}} $$

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \(\frac{z^{2} + 12z + 20}{z - 1}\).

Step 1: Factorise the numerator and denominator

Factorise the numerator \[3 z^{4}+36 z^{3}+60 z^{2}\] as \(3z^{2}(z^{2} + 12z + 20)\) and the denominator \[3 z^{3}-3 z^{2}\] as \(3z^{2}(z - 1)\). This enables us to see any common factors in the numerator and denominator.

Step 2: Cancel common factors

Having factorised both the numerator and denominator, we can see that they each have a \(3z^{2}\) term. Divide both numerator and denominator by this common factor to simplify the expression to \(\frac{z^{2} + 12z + 20}{z - 1}\).

Step 3: Check if further simplification is possible

After the previous step, there are no common factors left in the numerator and denominator. So we can conclude that the simplified form of the given fractional expression is \(\frac{z^{2} + 12z + 20}{z - 1}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the building blocks of a polynomial by breaking it down. This process is crucial when simplifying rational expressions. The task often involves recognizing common terms or using special formulas such as the difference of squares. In our example, the numerator is represented by the polynomial \(3z^4 + 36z^3 + 60z^2\). Here, you notice that all terms share a factor of \(3z^2\). Factoring this out gives us \(3z^2(z^2 + 12z + 20)\). This step makes it easier to identify and cancel common factors later. The denominator \(3z^3 - 3z^2\) is also factorable using the common factor \(3z^2\) again, resulting in \(3z^2(z - 1)\). Both factorizations reveal the simple underlying polynomials, paving the way for cancellation of common factors.

Canceling Common Factors

After factoring, the next step is to cancel out any common factors in the numerator and denominator. This is crucial because it simplifies the rational expression, making it more manageable. In our case, the factor \(3z^2\) appears in both the numerator and the denominator following the factorization.

• Identify the common factors in both parts of the fraction.
• Cancel these factors by dividing both the numerator and numerator by them.

This simplification leaves us with \(\frac{z^2 + 12z + 20}{z - 1}\). By canceling, we reduce complexity and make it easier to understand or further manipulate the expression if needed.

Rational Expressions

Rational expressions behave like fractions and involve polynomials in their numerators and denominators. Understanding how to manipulate these expressions is essential in algebra. The goal is often to simplify them as much as possible. Simplification can involve factoring, canceling common factors, and sometimes rewriting the expression in a different form. In the exercise, the given rational expression was \( \frac{3z^4 + 36z^3 + 60z^2}{3z^3 - 3z^2} \).

• Start by factoring both the numerator and the denominator to their simplest forms.
• Look out for common terms that can be canceled to streamline the expression.
• Check if the simplified rational expression has reached its simplest form or if further reduction is possible. If not, you're done!

This approach not only aids in better representation but also prepares you for solving further equations or evaluating expressions effectively.