Problem 32 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Find the quotient. $$ \frac{x^2-5 x-36}{x+2} \div\left(x^2-18 x+81\right) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{(x + 4)}{(x + 2)(x - 9)}\)

Step 1: Factorization

Firstly, factorize each of the polynomial expressions. For \(x^2 - 5x - 36\), the factors are \(x - 9\) and \(x + 4\). The expression \(x^2 - 18x + 81\) is a perfect square trinomial, so it can be written as \((x - 9)^2\). The given expression will look like this when factorized: \[\frac{(x - 9)(x + 4)}{x + 2} \div (x - 9)^2\]

Step 2: Simplify Division

Rewrite the division as multiplication with the reciprocal. The expression becomes \[\frac{(x - 9)(x + 4)}{x + 2} \cdot \frac{1}{(x - 9)^2}\]

Step 3: Simplifying the Expression

In the multiplication, \(x - 9\) cancels out with one of the \(x - 9\) terms on the right side of multiplication to leave \(\frac{(x + 4)}{(x + 2)(x - 9)}\). This is the final simplified form of the given expression.

Key Concepts

Factorization

Factorization involves breaking down a complex expression into simpler components, often in the form of factors. Factorization is particularly useful as it simplifies the process of solving equations or simplifying complex expressions. Let's take the example from our exercise.For the polynomial expression \(x^2 - 5x - 36\), we need to find two numbers that multiply to \(-36\) and add up to \(-5\). These two numbers are \(-9\) and \(+4\), which means the expression can be factorized into \((x - 9)(x + 4)\). As for \(x^2 - 18x + 81\), it is a perfect square trinomial. This means it can be rewritten as \((x - 9)^2\) since \(x - 9\) is repeated as the factor. By breaking down the polynomials into their factors, we simplify the expressions, making it easier to apply further operations, such as division.

Simplifying Expressions

Simplifying expressions is about making them easier to understand or use by reducing their complexity while maintaining their value. In the context of polynomial division, simplifying often involves rewriting the expression in a new form where similar terms or factors can be cancelled. The exercise involves rewriting the division as a multiplication with a reciprocal, an essential step in simplifying complex expressions. By turning the division into a multiplication, we get \(\frac{(x - 9)(x + 4)}{x + 2} \cdot \frac{1}{(x - 9)^2}\). This allows us to directly compare and cancel like factors across the entire expression. By cancelling \((x - 9)\), which appears both in the numerator and denominator, we arrive at the simplified form, \(\frac{(x + 4)}{(x + 2)(x - 9)}\). Thus, gridlocked complexities are removed, leading to clearer and more manageable expressions.

Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator and the denominator. Just like regular fractions, these can also be simplified, added, subtracted, multiplied, or divided. In our example, the rational expression initially presented involves dividing two polynomial fractions: \(\frac{x^2-5x-36}{x+2}\) by \(x^2-18x+81\). Understanding rational expressions is crucial because it allows us to manipulate and simplify complex algebraic fractions.After factorization, we rewrite the expression as a rational expression division, which was then turned into multiplication by the reciprocal, which gave us \(\frac{(x - 9)(x + 4)}{x + 2} \cdot \frac{1}{(x - 9)^2}\). This transformation made it easier to see which terms could be cancelled, simplifying to \(\frac{(x + 4)}{(x + 2)(x - 9)}\). Working with rational expressions often involves recognizing common factors and simplifying them to make the expressions manageable. This understanding is beneficial in solving real-life problems involving ratios and proportions. The key is to simplify without changing the expression's inherent value.