Problem 11 - College Algebra

Question

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$ \frac{y^{2}+7 y-18}{y^{2}-3 y+2} $$

Step-by-Step Solution

Answer

The simplified rational expression is \(\frac{y+9}{y-1}\) where \(y \neq 1\)

Step 1: Factoring the numerator and denominator

Factor the given rational expression. If the numerator and denominator can be factored, simplify the rational expression by factoring. For the numerator, \(y^{2}+7 y-18\), factors into \((y-2)(y+9)\). Similarly, the factored form of the denominator \(y^{2}-3 y+2\) is \((y-2)(y-1)\).

Step 2: Cancelling common factors

Look for common factors in the numerator and denominator. Here, \(y-2\) is the common factor. When we cancel the common factor, the rational expression simplifies to \(\frac{y+9}{y-1}\).

Step 3: Find the values that must be excluded

A rational expression is undefined when its denominator equals to zero. So set the denominator equal to zero and solve for \(y\). The solution \(y=1\) must be excluded from the domain as it would make the denominator zero.

Step 4: Write the final answer

The simplified rational expression is \(\frac{y+9}{y-1}\) where \(y \neq 1\)

Key Concepts

Factorization of Polynomials

Polynomial factorization is a process that breaks down a polynomial into simpler 'factored' forms, consisting of products of factors that are themselves polynomials. It is akin to factoring numbers, such as breaking down 15 into 3 times 5.

Consider the polynomial in the numerator of our exercise, \(y^{2}+7y-18\). To factor it, we search for two numbers that multiply to give -18 (the constant term) and add to give 7 (the coefficient of \(y\)). The numbers 9 and -2 fit the bill, so our polynomial factors into \((y-2)(y+9)\).

In the same manner, the denominator, \(y^{2}-3y+2\), factors into \((y-2)(y-1)\), since -2 and -1 multiply to 2 and add to -3. Correctly factoring polynomials is crucial for simplifying rational expressions. It allows us to see which terms can potentially be cancelled out, thus leading to a much simpler form.

Finding Domain of Rational Expressions

The domain of a rational expression is the set of all possible values that the variable can take without causing the denominator to be zero. Remember, division by zero is undefined and not allowed in mathematics.

When simplifying rational expressions, identifying the restrictions on the variable is just as important as the process of simplification itself. After the simplification, it is essential to recall the original denominator before simplification to exclude all values that would make any denominator zero.

In our exercise, we simplified to \(\frac{y+9}{y-1}\), which would be undefined at \(y=1\). However, the original denominator was \(y^{2}-3y+2\). To find all exclusions for the domain, we set each factor of the original denominator, \((y-2)(y-1)\), equal to zero. Thus, the values \(y=2\) and \(y=1\) must be excluded from the domain, even though \(y=2\) is no longer visible in the simplified form.

Canceling Common Factors

Once a rational expression is factored, any common factor present in both the numerator and the denominator can be cancelled out. This process is similar to simplifying fractions such as \(\frac{4}{8}\) to \(\frac{1}{2}\) by cancelling the common factor of 4.

In our example, after factoring, we notice \((y-2)\) appearing in both the numerator and the denominator. Cancelling this common factor gives us the simplified expression \(\frac{y+9}{y-1}\). Crucial to this step is understanding that cancelling is only valid when the terms are multiplication factors; we cannot cancel terms that are added or subtracted.

Always ensure you are cancelling only factors and not terms that are not part of a product. This is why factoring is a mandatory first step as it often reveals the multiplicative relationship between terms that addition or subtraction might hide.