Problem 62 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Question: Multiply the monomial \(9y^3\) with the binomial \((3y^2+2)\) and simplify the result. Answer: The result of multiplying the monomial with the binomial and simplifying is \(27y^5 + 18y^3\).

Step 1: Distribute the monomial

Multiply the monomial \(9y^3\) by each term of the binomial \((3y^2+2)\): $$ 9y^3(3y^2) + 9y^3(2) $$

Step 2: Perform the multiplications

Now, multiply the coefficients and the variables separately for each term: $$ (9 \cdot 3)y^{3+2} + (9 \cdot 2)y^3 $$

Step 3: Simplify the terms

Simplify the coefficients and the exponents: $$ 27y^5 + 18y^3 $$ There are no like terms to combine. So, the final answer is: $$ 27y^5 + 18y^3 $$

Key Concepts

Distribute the Monomial

When multiplying polynomials, it's essential to distribute the monomial correctly. This process involves taking a single term, the monomial, and multiplying it by every term in the polynomial it's being multiplied with. Let's consider our monomial, which is the term outside of the parenthesis, and multiply it with each term inside the parenthesis. For our example, we have a monomial, \(9y^3\), and a binomial, \(3y^2+2\). We distribute as follows:

\(9y^3\) is multiplied with \(3y^2\) and also with \(+2\), giving us two separate products. This means you write out separately \(9y^3\) times \(3y^2\) and \(9y^3\) times \(+2\), leading to \(9y^3(3y^2) + 9y^3(2)\). Make sure to multiply the monomial with each term of the polynomial one by one to avoid any mistakes.

Combine Like Terms

Once we have distributed the monomial and have a set of terms, the next step in simplifying a polynomial expression is to combine like terms. Like terms are terms that have the exact same variable parts and exponents, meaning they can be added or subtracted from each other. However, in the provided example:

\(27y^5 + 18y^3\),
there are no like terms because the terms have different exponents on the variable \(y\). The term \(27y^5\) has \(y\) raised to the power of 5, while \(18y^3\) has \(y\) raised to the power of 3. These two cannot be combined further, so they remain as they are. It's always important to check your final expression to ensure that all like terms have been combined and the expression is as simple as possible.

Simplify Expressions

To simplify expressions in polynomial multiplication, especially after distributing the monomial, you need to follow certain steps systematically. We start by multiplying the coefficients (numerical parts) and then apply exponent rules to the variable parts. In our example, multiplication of coefficients looks like this:

\((9 \cdot 3)y^{3+2} + (9 \cdot 2)y^3\).
You'll get \(27y^5 + 18y^3\) after carrying out the multiplication. When simplifying, remember that simplification might include reducing fractions, combining like terms, and applying exponent rules. Since there are no like terms in our result, we've achieved the simplest form of this particular expression.

Exponent Rules

Exponent rules are crucial when dealing with polynomial multiplication. One of the fundamental rules is when multiplying powers with the same base, you add the exponents. This rule was applied in our example when we multiplied \(9y^3\) by \(3y^2\). Instead of multiplying \(y^3\) by \(y^2\) directly, we use the rule and add their exponents:

\(y^3 \cdot y^2 = y^{3+2} = y^5\).
This is why the first term became \(27y^5\). It's essential to remember these rules as they are the backbone of simplifying polynomial expressions. Other exponent rules include raising a power to a power, where you multiply exponents, and raising a product to a power, where you apply the exponent to each factor within the parentheses.