Problem 68 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 8 a^{3} b^{2} c\left(2 a b^{3}+3 b\right) $$

Step-by-Step Solution

Answer

Question: Multiply the given expression and combine like terms: \(8 a^{3} b^{2} c (2 a b^{3}+3 b)\). Answer: \(16 a^{4} b^{5} c + 24 a^{3} b^{3} c\)

Step 1: Identify the distributive property

The distributive property states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products of the number and each term in the sum (or difference). In this case, we have a monomial, \(8 a^{3} b^{2} c\), multiplied by a binomial, \((2 a b^{3} + 3 b)\).

Step 2: Apply the distributive property

To apply the distributive property, multiply the monomial by each term in the binomial, and then add the results: $$ 8 a^{3} b^{2} c (2 a b^{3}+3 b) = 8 a^{3} b^{2} c \cdot 2 a b^{3} + 8 a^{3} b^{2} c \cdot 3 b $$

Step 3: Multiply the terms

Now, multiply the terms together. Remember that when multiplying terms with the same base, we add their exponents: $$ 8 a^{3} b^{2} c \cdot 2 a b^{3} = 16 a^{4} b^{5} c $$ $$ 8 a^{3} b^{2} c \cdot 3 b = 24 a^{3} b^{3} c $$ Combine the two products: $$ 16 a^{4} b^{5} c + 24 a^{3} b^{3} c $$

Step 4: Check for like terms and simplify

Now, look for any like terms to combine. In this case, there are no like terms (the terms have different variable combinations and exponents), so the expression is already in its simplest form: $$ 16 a^{4} b^{5} c + 24 a^{3} b^{3} c $$ That's the final expression after performing the multiplications and combining like terms.

Key Concepts

Multiplying Polynomials

Multiplying polynomials is a critical skill in algebra that can be easily tackled by understanding the distributive property. To multiply a monomial by a polynomial, as seen in the exercise \(8 a^{3} b^{2} c(2 a b^{3}+3 b)\), we distribute the monomial to each term of the polynomial.

For each distribution, we multiply the coefficients (numerical parts) together and then handle the variables. If the variables are the same, we use the exponent rules to combine them. For instance, when multiplying \(a^{3}\) by \(a\), we add the exponents, resulting in \(a^{4}\). It's all about systematically applying these steps to each term in the polynomial.

It's important to proceed with care to avoid mistakes, especially in more complicated polynomials. After the multiplication, we end up with a new polynomial with terms that may or may not be like terms.

Combining Like Terms

Once we have finished multiplying polynomials, we may find ourselves with an expression that contains like terms. These are terms that have exactly the same variables raised to the same powers but may have different coefficients. In our example, \(16 a^{4} b^{5} c + 24 a^{3} b^{3} c\) does not contain like terms because the exponents on the variables differ.

However, if you encounter an expression where like terms are present after multiplying, they must be combined to simplify the expression. This is usually done by adding or subtracting the coefficients while keeping the variable part unchanged. Remember, only like terms can be combined; different terms with different variables or exponents are left as is.

Exponent Rules

Understanding exponent rules, also known as laws of exponents, is vital when working with polynomials. These rules tell us how to handle expressions with powers when we are multiplying, dividing, or raising them to another power.

For example, when multiplying variables with the same base, we add their exponents as shown in the step \(8 a^{3} b^{2} c \cdot 2 a b^{3} = 16 a^{4} b^{5} c\). It's essential to remember that exponents only apply to the base they are directly connected to and that these rules help keep the work with polynomials organized and manageable.

There are other exponent rules like the power of a product rule, power of a quotient rule, and the power of a power rule that govern different situations. These exponent rules simplify the process of manipulating expressions and are fundamental to understanding higher-level algebra.