Problem 96 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 2 a^{2}(a+4)(a+3) $$

Step-by-Step Solution

Answer

Answer: The result of multiplying the given polynomials is \(2a^4+14a^3+24a^2\).

Step 1: Identify the polynomials to be multiplied

We are given the product of three terms: \(2a^{2}\), \((a+4)\), and \((a+3)\). We will first multiply \((a+4)\) and \((a+3)\) and then multiply the result with \(2a^{2}\).

Step 2: Multiply \((a+4)\) and \((a+3)\)

Using the distributive property, multiply \((a+4)\) and \((a+3)\): $$(a+4)(a+3) = a(a+3)+4(a+3) = a^2+3a+4a+12$$ Now, combine the like terms: $$a^2+3a+4a+12 = a^2+7a+12$$

Step 3: Multiply the result with \(2a^{2}\)

Now, multiply the result from Step 2 with \(2a^{2}\): $$2a^{2}(a^2+7a+12) = 2a^{2}(a^2)+2a^{2}(7a)+2a^{2}(12) = 2a^4+14a^3+24a^2$$

Step 4: Write the final expression

The final simplified expression after performing the multiplication and combining like terms is: $$2a^4+14a^3+24a^2$$

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, particularly after multiplication, we often encounter similar elements that can be brought together—a process known as combining like terms. Like terms are terms that contain the same variables raised to the same power. For instance, in the exercise \(a^2 + 3a + 4a + 12\), the terms \(3a\) and \(4a\) are considered like terms because they each contain the variable \(a\), both to the first power.

To combine them, simply add their coefficients: \(3+4\) gives us \(7\), hence \(3a + 4a\) simplifies to \(7a\). This principle extends to more complex polynomials as well, where attention to the variables and their powers is crucial for accurate simplification.

Distributive Property

The distributive property is essential for multiplying polynomials. It’s a rule used to complete the multiplication across an expression enclosed within parentheses when it's being multiplied by another term or polynomial. As demonstrated in the given exercise, multiplying the polynomials \(a+4\) and \(a+3\) involves distributing each term of the first polynomial across all terms of the second.

Let's detail this step: taking \(a\) from the first polynomial and multiplying it by both \(a\) and \(3\) in the second gives us \(a^2 + 3a\). Similarly, multiplying the \(4\) by both terms of the second polynomial gives us \(4a + 12\). This property ensures that we 'distribute' the multiplication equally across all terms involved. The key is being methodical, ensuring each term is multiplied with every other term, before moving on to combining like terms as the next step.

Algebraic Expressions

In the realm of mathematics, algebraic expressions are combinations of letters and numbers representing a formula or a relationship between quantities. These expressions involve variables (like \(a\) in our exercise), constants (like the \(12\)), and an arrangement of algebraic operations (such as addition, subtraction, multiplication, division, exponents, etc.).

An important aspect of these expressions is their versatility and applicability to various mathematical problems, from simple arithmetic to complex calculus. Understanding the underlying structure and rules, such as the distributive property and how to combine like terms, is crucial for successfully manipulating these expressions to solve equations and inequalities.