Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Factor the trinomial.\(x^{2}-5 x-150\)

Step-by-Step Solution

Answer

The factored form of the given trinomial, \(x^{2}-5 x-150\), is \( (x - 15)(x + 10)\).

Step 1: Identify the coefficients and constant

First, identify the coefficients and constant in the trinomial. The coefficients are the numbers in front of the variables and the constant is the number without a variable. In this case, there is 1, which is the coefficient of \(x^{2}\), -5 which is the coefficient of \(x\), and -150 which is the constant.

Step 2: Find two numbers

We need to find two numbers that multiply to -150 and add up to -5. Knowing multiplication facts and using trial and error, we find that -15 and 10 satisfies these conditions because \(-15*10 = -150\) and \(-15+10 = -5\).

Step 3: Write the Factored Form

Use the two numbers to write the factored form of the trinomial. The factored form is \( (x - 15)(x + 10)\). Check the result by expanding, and it should return the original trinomial.

Key Concepts

Coefficients in Polynomials

In polynomial expressions, coefficients play a crucial role. They are the numbers that sit in front of the variables and determine the general shape and behavior of the polynomial when graphed. For example, in the trinomial \(x^2 - 5x - 150\), we have two coefficients:

• 1 for \(x^2\), often understood as "implicit" if it's not written,"
• -5 for \(x\).

These coefficients tell us how the terms in the polynomial scale with the variables. The constant term, such as -150 in this trinomial, represents the y-intercept when the polynomial is graphed. Understanding how constants and coefficients interact helps in the factoring process, as they guide us in finding relationships between the roots of the polynomial and its terms.

Factored Form

The factored form of a polynomial is essential in simplifying and solving polynomial equations. When a trinomial is expressed in factored form, the polynomial is broken into simplified algebraic products, making it easier to find the roots. The factored form of our example \(x^2 - 5x - 150\) is \((x - 15)(x + 10)\).
When you express a polynomial like this, it's equivalent to identifying the values of \(x\) that would make the original polynomial equal to zero. This form is especially useful in solving equations, as each factor set to zero gives us the solutions.

• For \((x - 15) = 0\), \(x = 15\) is a solution.
• For \((x + 10) = 0\), \(x = -10\) is another solution.

By using the factored form, you'll find the roots, which can help you understand the behavior of the polynomial through these critical points.

Trial and Error Method

The trial and error method is an invaluable strategy in math, especially for factoring trinomials when simple formulas aren't applicable. This tactic involves making educated guesses and tweaking them until you find a solution that works. In the context of factoring trinomials, your goal is to find two numbers that would multiply to give the product of the first term's coefficient and the constant term, and add to give the middle term's coefficient.
In our trinomial \(x^2 - 5x - 150\), you search for two numbers whose product is \(-150\) (product of the coefficient of \(x^2\) and the constant) and sum is \(-5\) (the coefficient of \(x\)). Through trial and error, arrive at -15 and 10:

• -15 * 10 = -150
• -15 + 10 = -5

By using trial and error, you can break down complex problems into more manageable parts, making it a useful technique when more straightforward methods don't clearly apply.