Problem 31 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. \(r^{2}+12 r+27\)

Step-by-Step Solution

Answer

The factored form of trinomial \(r^{2}+12 r+27\) is \((r + 9) (r + 3)\)

Step 1: Identify the trinomial

The trinomial given is \(r^{2}+12 r+27\)

Step 2: Determine the pair of numbers

Find two numbers that add up to 12 (the coefficient of the middle term) and multiply to 27 (the constant term). Upon inspection, these numbers will be 9 and 3, since \(9 + 3 = 12\) and \(9 * 3 = 27\)

Step 3: Write the factored form

Write the trinomial as a product of binomials using the pair of numbers determined before. The factored form will be \((r + 9) (r + 3)\)

Step 4: Check the factorization using FOIL

Confirm the factorization by multiplying the binomials using the FOIL method. That would be \(First: r*r = r^2\), \(Outer: r*3 = 3r\), \(Inner: 9*r = 9r\), \(Last: 9*3 = 27\). When you add those up, you get back to the original trinomial: \(r^2 + 3r + 9r + 27 = r^2 + 12r + 27\)

Key Concepts

FOIL Method

The FOIL method is a simple technique for multiplying two binomials. It’s an acronym that stands for: First, Outer, Inner, Last., which refers to the four pairs of terms you multiply together.

• First: Multiply the first terms in each binomial. For example, in \( (r+9)(r+3) \), you calculate \( r \times r = r^2 \).
• Outer: Multiply the outer terms. Here, it’s \( r \times 3 = 3r \).
• Inner: Next, multiply the inner terms. So, \( 9 \times r = 9r \).
• Last: Finally, multiply the last terms in each binomial, \( 9 \times 3 = 27 \).

Combine these results to verify the expression: \( r^2 + 3r + 9r + 27 = r^2 + 12r + 27 \).

This method helps check factorizations by ensuring multiplying back leads to the original expression.

Binomials

Binomials are algebraic expressions that consist of two terms separated by a plus or minus sign. In this context, we used binomials to rewrite the trinomial \( r^2 + 12r + 27 \).

Some key points about binomials include:

• Each term in a binomial can represent any real number or variable.
• When factoring trinomials like the one we have, we express them as a product of two binomials.
• In our example, \( (r + 9)(r + 3) \) is the factorization, which means these binomials multiply to give the original trinomial.

Understanding binomials is crucial for performing operations in algebra such as expansion and factoring.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They allow us to represent mathematical relationships in a general way.

In the exercise, we worked with a trinomial, a type of algebraic expression with three terms: \( r^2 + 12r + 27 \).

Features of algebraic expressions:

• They can include variables like \( r \), constants, and arithmetic operators (e.g., +, -).
• Factoring involves breaking down these expressions into simpler components, like binomials.
• Simplifying expressions makes it easier to solve equations or understand mathematical relationships.

Grasping the concept of algebraic expressions helps in various areas of algebra, making topics like factoring more intuitive.