Problem 22 - Intermediate Algebra

Question

Perform each division. \(\frac{5 y^{4}+45 y^{3}}{15 y^{2}}\)

Step-by-Step Solution

Answer

The result is \(\frac{1}{3}y^2 + 3y\).

Step 1: Simplify the expression

Start by simplifying the division \[\frac{5y^4 + 45y^3}{15y^2}.\]Each term in the numerator, \(5y^4\) and \(45y^3\), can be divided by the denominator \(15y^2\). Factor out the greatest common factor which is \(5y^2\).

Step 2: Factor the numerator

Factor \(5y^2\) out of the numerator to get: \[5y^2(y^2 + 9y).\]

Step 3: Cancel common factors

Now, rewrite the expression as: \[\frac{5y^2(y^2 + 9y)}{15y^2}.\]Cancel the common factor \(5y^2\) in both the numerator and the denominator: \[\frac{5y^2}{15y^2} = \frac{1}{3}.\] This leaves you with: \[\frac{1}{3}(y^2 + 9y).\]

Step 4: Distribute across the terms

Distribute \(\frac{1}{3}\) across \((y^2 + 9y)\), giving \[\frac{1}{3}y^2 + 3y.\]

Key Concepts

Algebraic Simplification

Algebraic simplification involves making mathematical expressions easier to work with. When dealing with polynomial division like \(\frac{5y^4 + 45y^3}{15y^2}\), the goal is to simplify by breaking down complex parts into simpler components.
Start by looking for a **greatest common factor** (GCF) that can be factored out of both the numerator and denominator. This helps reduce the expression to its simplest form.

• For the expression \(5y^4 + 45y^3\), the GCF is \(5y^2\). Factoring this out makes the expression clearer:
• It turns \(5y^4 + 45y^3\) into \(5y^2(y^2 + 9y)\).
• Following this, you divide each term by \(15y^2\), simplifying further to \(y^2 + 9y\).

By using algebraic simplification, complex expressions are reduced, ensuring equations or formulas are easier to compute and interpret. It’s a crucial skill for solving equations efficiently in algebra.

Factoring Polynomials

Factoring polynomials involves breaking them down into products of simpler polynomials, which is fundamental for solving polynomial equations or simplifying expressions.
In our solution, the polynomial \(5y^4 + 45y^3\) is factored by finding common elements. Identifying and extracting the greatest common factor simplifies the expression considerably.

• Consider the polynomial \(5y^4 + 45y^3\). The numbers 5 and 45 share a common factor of 5, and the powers of y, \(y^4\) and \(y^3\), share \(y^2\).
• Thus, by factoring out \(5y^2\), you get \(5y^2(y^2 + 9y)\).
• This step is crucial, as simplifying earlier rather than later avoids complex mathematical operations.

Factoring allows you to see relationships within expressions, such as how terms can be combined or divided efficiently. It’s about recognizing patterns or common elements.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Dividing polynomials results in these expressions, and simplifying them often involves reducing them to their simplest terms. The expression \(\frac{5y^2(y^2 + 9y)}{15y^2}\) is an example of a rational expression. To simplify:

• Identify and cancel out the common factors in both the numerator and the denominator.
• Here, \(5y^2\) is common in both, simplifying the expression to \(\frac{1}{3}(y^2 + 9y)\).
• Understanding rational expressions helps enable easier calculations and solving algebraic equations.

Rational expressions help algebraists handle operations on polynomials efficiently, much like numerical fractions but with polynomial rules applied.