Problem 57 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 3 x(5 x+4) $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((3x)(5x+4)\). Answer: The simplified expression is \(15x^2 + 12x\).

Step 1: Apply distributive property

To multiply \((3x)(5x+4)\), distribute \(3x\) to both terms in the parentheses \((5x + 4)\) as follows: $$ 3x(5x + 4) = 3x(5x) + 3x(4) $$

Step 2: Perform multiplication

Next, we will perform the multiplication to obtain: $$ 3x(5x) + 3x(4) = 15x^2 + 12x $$

Step 3: Combine like terms (if any)

In this case, there are no like terms to combine. So, the simplified expression is: $$ 15x^2 + 12x $$

Key Concepts

Distributive Property

The distributive property is a cornerstone concept in algebra, making it possible to simplify expressions by distributing a single term across terms within a parenthesis. When encountering an expression like \(3x(5x+4)\), the distributive property allows us to 'multiply out' the parentheses. This means we take the number or variable outside the parentheses—in this case, \(3x\)—and multiply it by each term inside the parentheses.

Let's break it down: You multiply \(3x\) by \(5x\), which yields \(15x^2\), and then multiply \(3x\) by \(4\), giving you \(12x\). These operations are performed separately, and the results are then added together to give the expanded expression \(15x^2 + 12x\).

• First multiplication: \(3x * 5x = 15x^2\)
• Second multiplication: \(3x * 4 = 12x\)

Remember, the distributive property is useful not only for polynomials but also in various algebraic expressions. It is essential in breaking down more complex problems into simpler steps.

Combining Like Terms

Once we have expanded an expression using the distributive property, we might often need to simplify the expression further by combining like terms. 'Like terms' are terms that have the exact same variables raised to the same powers.

In the expression from the previous process, \(15x^2 + 12x\), there are no like terms to combine since \(15x^2\) and \(12x\) have different powers of \(x\). But, if we had a case with a resulting expression such as \(15x^2 + 7x - 2x^2 + 3x\), we would look for terms with the same variable and exponent to combine.

• Combine \(15x^2\) and \(-2x^2\) to get \(13x^2\)
• Addition of \(7x\) and \(3x\) results in \(10x\)

Eventually, we would end up with \(13x^2 + 10x\) as the simplified expression. Combining like terms is a straightforward but powerful tool to tidy up algebraic expressions and polynomials.

Polynomial Multiplication

Polynomial multiplication might seem daunting, but it's a matter of systematically applying the distributive property over and over until every term is accounted for. When multiplying polynomials, you'll distribute each term of the first polynomial across all terms of the second. For instance, if multiplying a binomial by a binomial, you'll have four products to combine at the end.

Consider the two binomials \((3x + 2)(x + 4)\). First, you'll distribute \(3x\) across the terms in the second binomial yielding \(3x^2 + 12x\), and then distribute \(2\) similarly to get \(2x + 8\). What follows is combining like terms across all products:

• Multiply: \(3x*x\) and \(3x*4\)
• Multiply: \(2*x\) and \(2*4\)
• Final expression: \(3x^2 + 12x + 2x + 8\)
• Combining like terms: \(3x^2 + 14x + 8\)

Understanding polynomial multiplication allows tackling larger and more complex expressions with ease, paving the way to mastering algebra and higher-level math.