Problem 64 - Elementary Algebra

Question

Factor. Either factor out the greatest common factor, factor by grouping, use the guess and check method, or use the \(a c\) method. $$ 8 c^{2}+26 c+15 $$

Step-by-Step Solution

Answer

\( (4c + 3)(2c + 5) \)

Step 1: Identify the type of polynomial

We have a quadratic polynomial in the form of \( ax^2 + bx + c \), where \( a = 8 \), \( b = 26 \), and \( c = 15 \).

Step 2: Multiply \(a\) and \(c\)

Calculate the product of \( a \) and \( c \):\( 8 \times 15 = 120 \).

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

We need two numbers that multiply to 120 and add up to 26. These numbers are 6 and 20.This is determined because:\( 6 \times 20 = 120 \)and\( 6 + 20 = 26 \).

Step 4: Break down the middle term

Rewrite the original equation by breaking down the middle term:\( 8c^2 + 6c + 20c + 15 \).

Step 5: Factor by grouping

Factor by grouping the terms in pairs:\( (8c^2 + 6c) + (20c + 15) \).Factor out the greatest common factor in each pair:\( 2c(4c + 3) + 5(4c + 3) \).

Step 6: Factor out the common binomial

Factor out the common binomial factor \( (4c + 3) \):\( (4c + 3)(2c + 5) \).

Key Concepts

quadratic polynomial

A quadratic polynomial is a type of polynomial that has a degree of 2. This means the highest exponent of the variable is 2. Quadratic polynomials typically look like this: \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The constants \(b\) and \(c\) can be any number, including zero.

• The term \( ax^2 \) is called the quadratic term.
• The term \( bx \) is the linear term because it has a degree of 1.
• The term \( c \) is the constant term because it doesn't include a variable.

In the given exercise, the quadratic polynomial is \( 8c^2 + 26c + 15 \). Here, \( a = 8 \), \( b = 26 \), and \( c = 15 \). To factor this polynomial, we can use different methods such as factoring by grouping, using the greatest common factor, or the AC method.

factoring by grouping

Factoring by grouping is a helpful method when dealing with polynomials with four terms. The idea is to group terms in pairs and factor out the greatest common factor from each pair. This method is especially useful after breaking down the middle term in a quadratic polynomial.
Here's how to do it:

• Group the terms in pairs: \( (8c^2 + 6c) + (20c + 15) \).
• Factor out the greatest common factor from each pair: \( 2c(4c + 3) + 5(4c + 3) \).
• Notice that \( (4c + 3) \) is common in both terms, so factor it out: \( (4c + 3)(2c + 5) \).

This step allows you to write the quadratic polynomial as a product of two binomials. It's important to practice this method to become comfortable with identifying common factors and grouping terms efficiently.

greatest common factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. In polynomial factorization, it is the highest degree term that is common in all terms of the polynomial. Factoring out the GCF simplifies the polynomial and makes other factoring methods easier.
For example, consider the polynomial \( 8c^2 + 26c + 15 \). Each term is checked for common factors:

• \( 8c^2 \) has factors 1, 2, 4, 8, \( c \), and \( c^2 \).
• \( 26c \) has factors 1, 2, 13, 26, and \( c \).
• \( 15 \) has factors 1, 3, 5, and 15.

Since they do not share a common factor larger than 1, we focus on other methods like the AC method or grouping. However, always check for a GCF at the beginning, as it simplifies the polynomial and may sometimes reveal a simpler factoring method.

AC method

The AC method is a factoring technique used for quadratic polynomials of the form \( ax^2 + bx + c \). It is particularly useful for polynomials where standard factoring methods are challenging.
Here's how to apply the AC method step-by-step:

• Identify coefficients \(a\), \(b\), and \(c\) in the polynomial.
• Multiply \(a\) and \(c\) to get \(ac\).
• Find two numbers that multiply to \(ac\) and add up to \(b\).
• Break up the middle term using these two numbers.
• Factor by grouping the terms in pairs and simplify.

In the given exercise:

• \(a = 8\), \(b = 26\), \(c = 15\).
• Calculate \(ac = 8 \times 15 = 120\).
• Find numbers that multiply to 120 and add to 26: these are 6 and 20.
• Rewrite the polynomial: \( 8c^2 + 6c + 20c + 15 \).
• Factor by grouping: \( (8c^2 + 6c) + (20c + 15) = 2c(4c + 3) + 5(4c + 3) \).
• Factor out the common binomial: \( (4c + 3)(2c + 5) \).

The AC method simplifies complex problems step-by-step and ensures accuracy in factoring quadratic polynomials.