Problem 111 - Introductory Algebra for College Students

Question

Multiply: \((3 x+4)(3 x+1)\)

Step-by-Step Solution

Answer

The short answer after simplifying the expression is \(9x^2 + 15x + 4\).

Step 1: Apply the First term multiplication

Multiply the first terms in both binomials. Specifically, multiply \(3x\) in the first binomial by \(3x\) in the second binomial. This gives: \(3x * 3x = 9x^2\).

Step 2: Apply the Outer term multiplication

Multiply the outer terms in the expression. Specifically, multiply \(3x\) from the first binomial by \(1\) from the second binomial. This gives: \(3x * 1 = 3x\).

Step 3: Apply the Inner term multiplication

Multiply the inner terms. Specifically, multiply \(4\) from the first binomial by \(3x\) from the second binomial. This gives: \(4 * 3x = 12x\).

Step 4: Apply the Last term multiplication

And finally, multiply the last terms of both binomials. Specifically, multiply \(4\) from the first binomial and \(1\) from the second binomial. This gives: \(4 * 1 = 4\).

Step 5: Combine the results

Combine all of the results from steps 1 through 4: \(9x^2\) from the first term multiplication, \(3x\) from the outer term multiplication, \(12x\) from inner term multiplication, and \(4\) from the last term multiplication. Combine all of these to get your final answer: \(9x^2 + 3x + 12x + 4\)

Step 6: Simplify the expression

Combine like terms in the expression \(9x^2 + 3x + 12x + 4\) to simplify. Specifically, combine the \(3x\) and \(12x\) terms: \(9x^2 + 15x + 4\)

Key Concepts

Binomial Multiplication

Binomial multiplication involves multiplying together two algebraic expressions that each contain two terms. In this exercise, the expression to be multiplied is

• ><(3x + 4)(3x + 1)

The method commonly used for binomial multiplication is often called the "FOIL" method. This stands for First, Outer, Inner, and Last, referring to the terms within each binomial that you multiply together to get part of the overall product.
The steps are:

• First: Multiply the first terms of each binomial, resulting in \(3x \times 3x = 9x^2\).
• Outer: Multiply the outer terms, obtaining \(3x \times 1 = 3x\).
• Inner: Multiply the inner terms, yielding \(4 \times 3x = 12x\).
• Last: Multiply the last terms, leading to \(4 \times 1 = 4\).

When you add all the individual results together, you have the un-simplified expression \(9x^2 + 3x + 12x + 4\).

Algebraic Expressions

Algebraic expressions are collections of numbers, variables, and operations that represent a quantity. In the context of this exercise, each binomial is an algebraic expression.
A binomial consists of exactly two terms, so a good example is each part of (3x + 4) and (3x + 1).

• The terms in these binomials are either constants, like "4" or "1", or they are variable terms, like \(3x\), which represents a constant (3) multiplied by a variable (x).

Algebraic expressions may also include operations, such as addition, in this case, illustrated by the "+" between terms in each binomial.
Mastering algebraic expressions includes understanding how to manipulate and transform them, which is essential for success in tasks like multiplication and simplification.

Distributive Property

The distributive property is a fundamental property of numbers and is particularly useful in algebra to simplify expressions. It states that multiplying a sum by a number gives the same result as multiplying each addend separately by the number and then adding the products. Mathematically, it's expressed as:

• \(a(b + c) = ab + ac\)

This property is central to binomial multiplication, allowing you to tackle expressions like \((3x + 4)(3x + 1)\) by breaking them down into simpler parts. In this problem:

• We use the distributive property to distribute each term in the first binomial across each term in the second binomial.

This involves four multiplication operations, one for each pairing of terms (i.e., first, outer, inner, last) in the two binomials.
Understanding the distributive property enhances your ability to multiply and simplify not just binomials, but various types of algebraic expressions.