Problem 48 - Elementary Algebra

Question

Use the \(a c\) method to factor. Check the factoring. Identify any prime polynomials. $$ a^{2}-4 a-32 $$

Step-by-Step Solution

Answer

The factored form of \(a^2 - 4a - 32\) is \((a - 8)(a + 4)\). The polynomial is not prime.

Step 1: Identify coefficients

Identify the coefficients of the quadratic expression. For the expression \(a^2 - 4a - 32\), \(a = 1\), \(b = -4\), and \(c = -32\).

Step 2: Calculate the product of a and c

Calculate the product of \(a\) and \(c\). This is \(1 \times -32 = -32\).

Step 3: Find two numbers that multiply to ac and add to b

Find two numbers that multiply to \(-32\) and add to \(-4\). These numbers are \(-8\) and \(4\) because \(-8 \times 4 = -32\) and \(-8 + 4 = -4\).

Step 4: Rewrite the middle term using found numbers

Rewrite the expression \(a^2 - 4a - 32\) using \(-8\) and \(4\): \(a^2 - 8a + 4a - 32\).

Step 5: Factor by grouping

Group the terms and factor each group: \(a(a - 8) + 4(a - 8)\).

Step 6: Factor out the common binomial

Factor out the common binomial factor \((a - 8)\): \((a - 8)(a + 4)\).

Step 7: Check factoring

Expand \((a - 8)(a + 4)\) to check the factoring: \((a - 8)(a + 4) = a^2 + 4a - 8a - 32 = a^2 - 4a - 32\).

Step 8: Identify prime polynomial if applicable

Since the expression \((a - 8)(a + 4)\) can be factored, it is not a prime polynomial.

Key Concepts

ac method

The ac method is a systematic approach for factoring quadratic expressions, particularly when the coefficient of the quadratic term is not 1. This method breaks down a quadratic expression by multiplying the coefficient of the quadratic term by the constant term and finding the appropriate pair of factors that can be used to rewrite the middle term.
Here's how to use the ac method:

- Identify coefficients a, b, and c. In the expression we’re working on, we have: \(a = 1\), \(b = -4\), and \(c = -32\).

- Multiply a and c. This gives us a product of -32.

- Find two numbers that multiply to -32 and add to -4. These values are -8 and 4 since \(-8 \times 4 = -32\) and \(-8 + 4 = -4\).

- Rewrite the middle term using the two numbers found. This helps us to split the expression and make it easier to factor by grouping.

quadratic expressions

Quadratic expressions are algebraic expressions where the highest exponent of the variable is 2. They often take the form of \(ax^2 + bx + c\). Understanding these expressions is important for solving quadratic equations and factoring.
The quadratic expression we are dealing with is \(a^2 - 4a - 32\). Here,

- The quadratic term is \(a^2\)

- The linear term is \(-4a\)

- The constant term is \(-32\)

Recognizing how the terms are related sets the stage for using techniques like the ac method to factor them into simpler binomial forms.

factoring by grouping

Factoring by grouping involves rearranging the terms in a polynomial and grouping them in pairs so that each group has a common factor. This method is especially useful in quadratic expressions after we have applied the ac method.
Let's see how it works with our expression:

- From the steps above, we rewrite the expression \(a^2 - 4a - 32\) into \(a^2 - 8a + 4a - 32\).

- We then group the terms: \(a(a - 8) + 4(a - 8)\).

- Notice how \(a - 8\) is a common factor. So, we factor it out and get: \((a - 8)(a + 4)\).

prime polynomials

A prime polynomial cannot be factored into the product of two or more non-trivial polynomials in the set of integers. They are akin to prime numbers in arithmetic.
To determine if a polynomial is prime:

- Attempt to factor it using methods such as the ac method or factoring by grouping.

- If no factors are found, the polynomial is considered prime.

In our worked example, \(a^2 - 4a - 32\) is not a prime polynomial because we were able to factor it into \((a - 8)(a + 4)\).

coefficients in algebra

Coefficients are numerical or symbolic multipliers of variables in algebraic expressions. They play a crucial role in the structure and solving of quadratic expressions.
In our quadratic expression, \(a^2 - 4a - 32\), each term has a coefficient:

- The coefficient of the quadratic term \(a^2\) is 1.

- The coefficient of the linear term \(-4a\) is -4.

- The constant term \(-32\) has no variable and acts as the coefficient itself.

Identifying and understanding these coefficients allows us to apply the ac method efficiently and correctly factor quadratic expressions by grouping.