Problem 102 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 3 a^{2}\left(2 a^{3}-10 a^{2}-4 a+9\right) $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(6a^5 - 30a^4 - 12a^3 + 27a^2\).

Step 1: Distributive Property

Apply the distributive property to multiply \(3a^2\) by each term inside the parentheses. $$ 3a^2(2a^3) - 3a^2(10a^2) - 3a^2(4a) + 3a^2(9) $$

Step 2: Perform Multiplication

Multiply the coefficients and add the exponents of like bases in each term. $$ 6a^5 - 30a^4 - 12a^3 + 27a^2 $$

Step 3: Finished Expression

Since there are no like terms to combine, we have our final simplified expression: $$ 6a^5 - 30a^4 - 12a^3 + 27a^2 $$

Key Concepts

Distributive Property

Understanding the distributive property is essential when dealing with algebra. It allows you to multiply a single term by a group of terms within parentheses. The principle behind it is straightforward: you distribute the multiplication of the single term to each term inside the parentheses individually.

Think of it as sharing equally. If you have three friends and you give each one a slice of pizza, you're distributing one pizza among three people. Similarly, in algebra, if you have a term outside the parenthesis, you 'give' it to each term inside the parenthesis. In our exercise, the term outside the parenthesis is \(3a^2\), and it needs to be distributed across each term inside: \(2a^3\), \(-10a^2\), \(-4a\), and +9.

By doing this, you’re setting the stage for further simplification, making it easier to combine like terms and solve the expression.

Combining Like Terms

Once you've distributed, it's time to bring like terms together through 'combining like terms.' Imagine you're organizing a fruit bowl. You'd group apples with apples, oranges with oranges, right? In algebra, 'like terms' are terms that have the same variables raised to the same powers. Coefficients can be different, but the variables must match.

In our exercise, after distributing \(3a^2\) and multiplying, we don't find any terms that have the same variable with the same exponent, hence there are no 'like terms' to combine. If there were, we would add or subtract the coefficients of these like terms to simplify the expression.

Multiplication of Polynomials

When multiplying polynomials, it's like conducting a full-blown orchestra – every element plays a crucial role. Breaking down the term 'polynomial,' 'poly' means 'many' and 'nomial' means 'terms.' So a polynomial is an expression with many terms.

In algebra, multiplying polynomials involves distributing every term of the first polynomial to every term of the second polynomial. It's crucial to keep order and organization throughout this process. In the provided exercise, we multiplied a monomial \(3a^2\) by a polynomial to obtain a new polynomial. This technique is used as a building block for more complicated operations in algebra, such as factoring and solving polynomial equations.

Exponents in Algebra

Exponents can be thought of as repeated multiplication. If you see an exponent, it tells you how many times to multiply the base by itself. For example, \(a^3\) means \(a \times a \times a\). In algebra, when you're multiplying two exponential expressions that have the same base, you keep the base and add the exponents together.

This rule streamlines operations with polynomials significantly. Referring back to our example: when we multiplied \(3a^2 \times 2a^3\), we simply added the exponents of \(a\) to get \(3\times 2a^{2+3} = 6a^5\). Remember, this rule only applies to multiplication with the same base; for division, you subtract the exponents, and for raising a power to a power, you multiply the exponents.