Problem 93 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. One factor of \(x^{2}+x+20\) is \(x+5\)

Step-by-Step Solution

Answer

The statement 'One factor of \(x^{2}+x+20\) is \(x+5\)' is false. The polynomial \(x^{2}+x+20\) does not have \(x+5\) as a factor. The corrected statement is: There are no real factors of \(x^{2}+x+20\) other than 1 and itself.

Step 1: Substitute k into the Polynomial

Substitute \(x=-5\) into the polynomial. That is, evaluate \(f(-5)\) where \(f(x)\) is \(x^{2}+x+20\), which equals \((-5)^{2}+(-5)+20\)

Step 2: Simplify the Expression

\((-5)^{2}+(-5)+20 = 25-5+20 = 40

Step 3: Determine the Truth Value

Since \(f(-5) = 40\), which is not equal to zero, \(x + 5\) is not a factor of the polynomial \(x^{2}+x+20\). The statement is false.

Step 4: Correction of the False Statement

To make the statement true, one must find an actual factor of the polynomial. But since \(x^{2}+x+20\) does not have any real factors, the corrected statement could be: There are no real factors of \(x^{2}+x+20\) other than 1 and itself.

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables, coefficients, and exponents. They are used to model relationships and solve equations. Let’s look closer at the polynomial in our exercise:

• The given polynomial is: \( x^{2} + x + 20 \).
• This polynomial has three terms: \( x^{2} \), \( x \), and 20.
• Each term in the polynomial has a different degree. In this example, the degrees are 2, 1, and 0, respectively.

Polynomials can be classified by their number of terms, such as monomials, binomials, and trinomials. In our example, we have a trinomial because it consists of three terms. Understanding these basic properties helps when performing operations or factoring polynomials.

Truth Value

The truth value of a statement tells us whether it is true or false. To determine the truth value of the initial statement that \( x + 5 \) is a factor of \( x^{2} + x + 20 \), we evaluated the expression by substituting a particular value for \( x \).

• If substituting results in zero, \( x + 5 \) would indeed be a factor of the polynomial, making the statement true.
• If it doesn't result in zero, as in this exercise where \( f(-5) = 40 \), the statement is false.

In our case, since \( f(-5) eq 0 \), \( x + 5 \) is not a factor. Therefore, the truth value of the statement that \( x+5 \) is a factor is false. Correcting false statements by finding true factors or acknowledging the lack of them helps maintain accuracy in mathematical conclusions.

Substituting Values

Substituting values in polynomials is a method to test if a specific value satisfies the polynomial expression, particularly useful when determining factors. Here’s how it works:

• Substitute the particular value for \( x \): in this case, \( x = -5 \).
• Evaluate the polynomial: substitute \(-5\) into \( f(x) = x^{2} + x + 20 \).

When substituting, we first calculate \((-5)^2 = 25\), then \(-5\), and finally add 20. This results in 40, showing that substituting \(-5\) into the polynomial does not yield zero.
Substituting values is a straightforward yet powerful tool in algebra. It allows us to identify whether certain expressions are factors, roots, or solutions to equations. It's like plugging coordinates into a function to check if they lie on its graph. This approach is instrumental in many areas of mathematics.