Problem 28 - Beginning and Intermediate Algebra

Question

Divide. $$\frac{u^{2}-11 u+30}{u-5}$$

Step-by-Step Solution

Answer

The simplified expression is \(u - 6\).

Step 1: Factor the quadratic polynomial

The quadratic polynomial given is \(u^2 - 11u + 30\). To factor this, we should look for two numbers that multiply to 30 and add up to -11. In this case, the numbers are -5 and -6. So, we can factor the polynomial as \((u - 5)(u - 6)\).

Step 2: Substitute the factored polynomial

Now, let's substitute the factored polynomial into the given expression. This gives us the following expression: \[\frac{(u - 5)(u - 6)}{u - 5}\]

Step 3: Cancel out common terms

In the expression, we can see that the term \((u - 5)\) appears both in the numerator and the denominator. We can cancel out these common terms which results in the following simplified expression: \[\frac{(u - 6)}{1}\]

Step 4: Final answer

As there is no need to divide by 1, the final simplified expression is: \(u - 6\)

Key Concepts

Quadratic polynomial factoring

Quadratic polynomial factoring is a fundamental concept in algebra that helps to simplify expressions and solve equations. When you have a quadratic polynomial like \(u^2 - 11u + 30\), you aim to express it as a product of two binomials. The key is to find two numbers that multiply to the constant term (in this case, 30) and add up to the linear coefficient (here, -11).
In our example, the numbers are -5 and -6 because

• -5 multiplied by -6 gives 30,
• and when added together, they total -11.

Thus, we can rewrite \(u^2 - 11u + 30\) as \((u - 5)(u - 6)\). This factorization is crucial for simplifying the expression further. Remember, this step sets the stage for canceling common terms in polynomial division.

Canceling common terms

When dividing polynomials, you often encounter common terms in the numerator and the denominator. These are terms that appear in both parts of the fraction and can be "canceled" to simplify the expression. By canceling, we mean effectively reducing the expression by eliminating these common factors.
For instance, consider the expression \(\frac{(u - 5)(u - 6)}{u - 5}\). Here, \(u - 5\) is a common factor present in both the numerator and the denominator. Since anything divided by itself equals one, the expression reduces to \(\frac{(u - 6)}{1}\).
Canceling common terms simplifies the process of polynomial division greatly and transforms complex expressions into more manageable forms. It highlights that you only multiply and divide terms, not add or subtract, while canceling.

Simplifying expressions

Simplifying expressions is a technique aimed at making expressions easier to understand and solve. It involves reducing expressions to their simplest form, eliminating unnecessary terms to clarify the structure of the expression. In our example, after factoring and canceling common terms, we end up with \(u - 6\).
This final expression is straightforward because it's reduced to a basic linear equation. Simplifying saves time and effort, especially when working with larger or more complex algebraic problems, by streamlining computations and avoiding cumbersome fractions.
The key is to remember the operations that led to the simplified form and ensure that they follow algebraic rules and logic. This careful approach prevents errors and helps in solving further algebraic equations efficiently.