Problem 6 - Algebra 2

Question

Simplify each expression. \(\frac{9 y^{2}-6 y^{3}}{2 y^{2}+5 y-12}\)

Step-by-Step Solution

Answer

\(\frac{3y^2(3-2y)}{(2y-3)(y+4)}\) is the simplified form.

Step 1: Factoring the Numerator

Let's start by factoring the numerator of the fraction, \(9y^2 - 6y^3\). We can factor out a common factor from each term. The common factor is \(3y^2\). Thus, \(9y^2 - 6y^3 = 3y^2(3 - 2y)\).

Step 2: Factoring the Denominator

Next, we need to factor the denominator, \(2y^2 + 5y - 12\). We look for two numbers that multiply to \(-24\) (the product of the coefficient of \(y^2\), which is 2, and the constant term, which is -12) and add up to \(5\). These numbers are \(8\) and \(-3\). Therefore, the expression can be factored as follows: \(2y^2 + 5y - 12 = (2y - 3)(y + 4)\).

Step 3: Simplifying the Expression

Now that both the numerator and the denominator are factored, substitute them back into the original expression: \[\frac{3y^2(3 - 2y)}{(2y - 3)(y + 4)}\]. There are no common factors between the numerator and the denominator that can be canceled out, so this is the simplified form of the expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is like solving a puzzle by breaking down expressions into simpler components. In this context, it helps us simplify algebraic expressions, typically making them easier to work with. When you factor a polynomial, you're looking for terms that can be grouped together as a common multiplier in the original equation. The goal is to rewrite the polynomial in a product form.

For example, in the numerator of our expression:

• The polynomial is \(9y^2 - 6y^3\).
• This can be factored by finding the greatest common factor (GCF), which in this case is \(3y^2\).

Thus, the factored form is \(3y^2(3 - 2y)\). Practicing by identifying the GCF in a polynomial and factoring out this common factor will make you proficient in polynomial factorizations.

Numerator and Denominator

In a fraction, the numerator and the denominator are two key components that determine its value. The numerator is the top part, representing how many parts of a whole we have. The denominator, on the bottom, tells us how many equal parts make up a whole.

In the expression from the exercise:

• The numerator is \(9y^2 - 6y^3\), the part that we simplified by factoring.
• The denominator is \(2y^2 + 5y - 12\), which also needs to be factored to simplify the fraction.

When simplifying fractions, both components must be handled carefully. The goal is to factor both sides and check if there are any common factors that can be canceled out, streamlining the fraction further.

Common Factor

The common factor is a crucial concept in algebra that allows us to simplify expressions by reducing them as much as possible. A common factor refers to a number or expression that divides two or more numbers or expressions without leaving a remainder. Identifying common factors is the first step in the factoring process.

Here's how it works in our solution:

• For \(9y^2 - 6y^3\), the common factor is \(3y^2\).
• This means both terms of the expression can be divided by \(3y^2\) to simplify it.

Once factored, you explore whether the new terms in the numerator and denominator share any common factors. If found, these can be "canceled" to further simplify the expression. That's why identifying and understanding common factors is essential for working with polynomials and fractions!