Problem 63 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$n^{2}-17 n+60$$

Step-by-Step Solution

Answer

\((n - 12)(n - 5)\) is the factorization of the polynomial.

Step 1: Identifying the Form

The polynomial is given as \(n^2 - 17n + 60\). This is a quadratic polynomial in the form of \(ax^2 + bx + c\), where \(a = 1\), \(b = -17\), and \(c = 60\). Our goal is to factor the quadratic into two binomials.

Step 2: Finding Two Numbers

To factor the quadratic, we need two numbers that multiply to \(c = 60\) and add up to \(b = -17\). We will list the pairs of factors of 60 and find which pair sums to -17.

Step 3: Testing Factor Pairs

The factor pairs of 60 are (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), and (6, 10). By checking the sums: \((4, 15)\) gives us \(4 + 15 = 19\), and \((-4, -15)\) gives us \(-4 + (-15) = -19\), none of these work for -17. Testing for sum: \(-12, -5\) gives us \(-12 + (-5) = -17.\) So, -12 and -5 are the correct pair.

Step 4: Writing the Binomial Factors

We can now express the factored form using the found pair: \((n - 12)(n - 5)\). These binomials multiply back to give the original quadratic polynomial, verifying they are indeed the correct factors.

Key Concepts

Quadratic Expressions

When dealing with polynomial expressions, quadratic expressions are one of the most common types. A quadratic expression is a polynomial of degree two, which means it has the highest power of the variable as square. A typical quadratic expression looks like this: \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

Some important points about quadratic expressions:

• They always form a parabola when graphed.
• The term \(bx\) represents the linear component, while the term \(c\) is the constant or y-intercept.
• Quadratic expressions are often factored to solve equations or to simplify them.

In the exercise provided, the quadratic expression is \(n^2 - 17n + 60\). It's important to identify this format because it sets the stage for factoring. Recognizing what \(a\), \(b\), and \(c\) are helps us begin the factoring process.

Binomial Factors

Once a quadratic expression is identified, the next step in factoring it is finding its binomial factors. Binomial factors are two simpler expressions that, when multiplied together, give back the original quadratic expression. In simpler terms, they are two-part polynomial expressions that break down the original into components that are easier to handle.

For instance, consider the expression \(n^2 - 17n + 60\). Our goal is to express this quadratic as a product of two binomials, typically written as \((n + m)(n + p)\). Determining \(m\) and \(p\) involves ensuring that:

• Their product is the constant term \(c\) (in this case, 60).
• Their sum is the linear coefficient \(b\) (here, -17).

By using these conditions, we find \(m = -12\) and \(p = -5\), leading us to the binomial factors \((n - 12)(n - 5)\). Checking and finding these factors is key to confirming the integrity of the solution.

Factoring Techniques

Factoring is a crucial skill in algebra, specifically when working with quadratic expressions. Several techniques are used, but the process often begins with identifying the form of the expression and using specific strategies to break it down into binomial factors.

*Key Techniques:*

• **Identifying Patterns**: Recognizing whether the quadratic is a perfect square or if it takes the form of a difference of squares can save time.
• **Trial and Error**: Listing factor pairs of the constant term and testing which pair adds up to the middle term is commonplace, as seen in the example.
• **Splitting the Middle Term**: This involves breaking down the linear component into two terms that match the factors, allowing grouping and simplifying the expression.

In our example of \(n^2 - 17n + 60\), using the trial and error technique to find two numbers that multiply to 60 and add to -17 is how we determined the factors \(-12\) and \(-5\). This approach is efficient for simple quadratics and ensures accurate results without overly complex calculations.