Problem 63 - Intermediate Algebra

Question

Divide, and then simplify, if possible. See Example 8. $$ \frac{y^{3}-9 y}{y+2} \div(y-3) $$

Step-by-Step Solution

Answer

\(\frac{y(y + 3)}{y + 2}\) is the simplified expression after division.

Step 1: Identify the Division of Fractions

When dividing two expressions, particularly when fractions are involved, recall that dividing by a number is the same as multiplying by its reciprocal. Thus, \( \frac{y^{3}-9y}{y+2} \div (y-3) \) can be rewritten as \( \frac{y^{3}-9y}{y+2} \times \frac{1}{y-3} \).

Step 2: Simplify the Expression \(y^3 - 9y\)

Factor the expression \(y^3 - 9y\) by taking out the common factor, which is \(y\). This gives \(y(y^2 - 9)\). Recognize \(y^2 - 9\) as a difference of squares, so it can be factored further: \((y - 3)(y + 3)\). Thus, \(y^3 - 9y = y(y - 3)(y + 3)\).

Step 3: Substitute Factored Expressions

Substitute the factorized form back into the expression. Now the expression is: \(\frac{y(y - 3)(y + 3)}{y + 2} \times \frac{1}{y - 3}\).

Step 4: Cancel Common Factors

Notice that \(y - 3\) appears in both the numerator and the denominator. Cancel the \(y - 3\) terms: \(\frac{y(y + 3)}{y + 2} \times \frac{1}{1}\).

Step 5: Final Simplified Expression

After canceling, multiply what's left: \(\frac{y(y + 3)}{y + 2}\). This is the simplest form of the expression after division. There aren't any further factors to cancel.

Key Concepts

Factoring Polynomials

When working with polynomials, factoring is a powerful technique to simplify expressions. Factoring reduces complex polynomials into their simplest components. For the polynomial \( y^3 - 9y \), the first step is to identify common factors. Here, \( y \) is a common factor across terms, so it's pulled out: \( y(y^2 - 9) \).
But the process isn't done there. Next, recognize any notable patterns such as the difference of squares. In \( y^2 - 9 \), we see a difference of squares since it can be written as \( y^2 - 3^2 \). This pattern is factored as \((y - 3)(y + 3)\).
By factoring completely, \( y^3 - 9y \) transforms into \( y(y - 3)(y + 3) \). This makes simplifying further steps easier, as seen in the solution. Factoring not only helps in simplifying polynomials but also in solving equations and finding roots.

Simplifying Expressions

Simplifying expressions often means reducing them to a more manageable form. For polynomial expressions like \( \frac{y^3 - 9y}{y + 2} \times \frac{1}{y - 3} \), simplification involves making use of factors and cancelling out terms.
Initially, after factoring \( y^3 - 9y \), the expression becomes \( \frac{y(y - 3)(y + 3)}{y + 2} \times \frac{1}{y - 3} \). Here, another step of simplification involves cancelling common factors between the numerator and the denominator, simplifying the task.
The term \( y - 3 \) exists in both the numerator and the denominator. By cancelling those out, the simplified expression is left as \( \frac{y(y + 3)}{y + 2} \). Simplification makes the expression less complex, helping achieve the simplest possible form, which is crucial for easier manipulation and understanding of the expression.

Difference of Squares

The difference of squares is a special type of factoring that follows the formula: \( a^2 - b^2 = (a - b)(a + b) \). This technique helps break down specific quadratic expressions into more straightforward factors easily.
In the exercise, the expression \( y^2 - 9 \) is a classic example of a difference of squares. Here, \( a = y \) and \( b = 3 \), which means \( y^2 - 9 \) factors to \( (y - 3)(y + 3) \).
Being aware of the difference of squares pattern equips you with a toolkit to simplify and factor expressions quickly. Recognizing such patterns can be immensely helpful as it cuts down on complex calculations, streamlining the solving process in problems involving polynomial division and simplification.