Problem 50 - Elementary Algebra

Question

Use the \(a c\) method to factor. Check the factoring. Identify any prime polynomials. $$ a^{2}+7 a-18 $$

Step-by-Step Solution

Answer

(a + 9)(a - 2)

Step 1: Write down the polynomial

Start with the given quadratic polynomial: a^2 + 7a - 18

Step 2: Identify a, b, and c

In the quadratic equation form ax^2 + bx + c, identify the coefficients a, b, and c. Here, a = 1, b = 7, and c = -18.

Step 3: Calculate the product of a and c

Multiply a and c: 1 × (-18) = -18

Step 4: Find two numbers that multiply to ac and add to b

Find two numbers that multiply to -18 and add up to 7. These numbers are 9 and -2 (since 9 * -2 = -18 and 9 + (-2) = 7).

Step 5: Rewrite the middle term

Rewrite the polynomial by breaking the middle term into two terms using the numbers found in Step 4: a^2 + 9a - 2a - 18

Step 6: Group the terms

Group the terms into two pairs: (a^2 + 9a) + (-2a - 18)

Step 7: Factor out the greatest common factor from each group

Factor out the greatest common factor from each pair: a(a + 9) - 2(a + 9)

Step 8: Factor the common binomial

Factor out the common binomial (a + 9): (a + 9)(a - 2)

Step 9: Check the factoring

To check the factoring, expand the binomials to ensure the original polynomial is obtained: (a + 9)(a - 2) = a^2 - 2a + 9a - 18 = a^2 + 7a - 18. Since this matches the original polynomial, the factoring is correct.

Step 10: Identify any prime polynomials

Since the polynomial can be factored into (a + 9)(a - 2), and both factors are linear, the given polynomial is not a prime polynomial.

Key Concepts

ac method

The ac method is a helpful technique for factoring quadratic polynomials. It’s named for the product of the coefficients a and c in the standard quadratic form, which is ax^2 + bx + c. To use the ac method:

• First, identify the coefficients a, b, and c in your quadratic polynomial.
• Next, multiply a and c together.
• Then, find two numbers that multiply to ac and add to b. These two numbers will help in rewriting the middle term.
• Break the middle term into two parts using the numbers from the previous step.
• Group the terms into two pairs and factor each group.
• Finally, factor out the common binomial to complete the factoring.

This method is systematic and works well for quadratics that can be factored. Practice helps in becoming proficient with the ac method.

quadratic polynomial

A quadratic polynomial is any polynomial of degree two. It has the general form: ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The quadratic polynomial will always graph into a parabolic shape that opens upwards if a > 0 and downwards if a < 0.

• The highest exponent in a quadratic polynomial is two.
• The term with the exponent of two is called the quadratic term.
• The term with the exponent of one is called the linear term.
• The constant term is the term without a variable attached.

Understanding the structure of a quadratic polynomial is key to using various factoring techniques, including the ac method, effectively.

greatest common factor

The greatest common factor (GCF) is the largest factor common to two or more terms. When factoring polynomials, finding the GCF simplifies the expressions and makes further factoring easier. To find the GCF:

• List the factors of each term.
• Identify the common factors.
• Select the largest common factor.

For example, in the polynomial a^2 + 9a - 2a - 18 from the exercise, the GCF of the terms in the group (a^2 + 9a) is a, and the GCF of the terms in the group (-2a - 18) is -2. Factoring these out helps simplify each group before factoring further.

factoring binomials

Factoring binomials is a crucial skill when working with quadratic polynomials. Binomials are simple polynomials with two terms. The goal when factoring is to express the binomial as a product of two or more factors. In the exercise, after grouping and factoring out the GCF, we find common binomials: a(a + 9) - 2(a + 9).
Here’s how you approach factoring binomials:

• Look for a common factor in both terms.
• Once identified, factor out the common binomial.

In our example, the common binomial is (a + 9), so we factor it out to get (a + 9)(a - 2). This method is useful in simplifying and solving quadratic polynomials.