Problem 17 - Elementary Algebra

Question

Multiply. (Assume all denominators are nonzero.) $$ 9+x 3 x+1 \cdot 3 x+9 $$

Step-by-Step Solution

Answer

The product is \(3x^2 + 28x + 9\).

Step 1: Expand the expression

Write the expression to be multiplied: \[(9 + x)(3x + 1) \]We want to multiply each term in the first expression by every term in the second expression.

Step 2: Distribute each term

Apply the distributive property:\[(9 + x)(3x + 1) = 9(3x) + 9(1) + x(3x) + x(1)\]This gives us the terms: \(27x\), \(9\), \(3x^2\), and \(x\).

Step 3: Combine like terms

Add together like terms from the expression:\[27x + 9 + 3x^2 + x\]Combine \(27x\) and \(x\) together to get \(28x\).

Step 4: Write the final expression

The fully simplified expression is:\[3x^2 + 28x + 9\]

Key Concepts

Distributive Property

The distributive property is a fundamental tool in algebra. It allows you to multiply a single term by two or more terms inside a parenthesis. When using this property, you effectively "distribute" the multiplication across each term within the parenthesis.
Consider the expression \(a(b + c)\). The distributive property allows us to expand this expression to \((ab + ac)\). In the original exercise, we have two binomials, \(9+x\) and \(3x+1\). By applying the distributive property, we multiply each term in the first binomial by each term in the second binomial.

• Multiply \(9\) by \(3x\), yielding \(27x\).
• Multiply \(9\) by \(1\), yielding \(9\).
• Multiply \(x\) by \(3x\), yielding \(3x^2\).
• Multiply \(x\) by \(1\), yielding \(x\).

This step-by-step application of the distributive property results in an expanded expression: \[27x + 9 + 3x^2 + x\] This method helps break down complex problems into manageable parts.

Combining Like Terms

After expanding our polynomial through the distributive property, the next crucial step is combining like terms. This process simplifies expressions by grouping and adding together the terms that have the same variables raised to the same power.
In the expanded expression, \[27x + 9 + 3x^2 + x\] observe that the terms \(27x\) and \(x\) are like terms. They both contain the variable \(x\) raised to the first power.

• Add \(27x\) and \(x\) together to get \(28x\).

After combining these like terms, the intermediate expression becomes \[3x^2 + 28x + 9\] This consolidation step is essential because it reduces the number of terms in the expression, making it much simpler and easier to interpret.

Simplifying Expressions

Simplifying expressions is the final step that brings everything together. This process concludes the task of turning a more complicated polynomial into a cleaner, more understandable form.
In our current polynomial \[3x^2 + 28x + 9\] all terms are combined and simplified. Simplifying generally involves performing any necessary arithmetic and combining like terms, which we have already done in prior steps.
The expression \[3x^2 + 28x + 9\] is written in standard polynomial form, starting with the highest degree term, \(3x^2\).
Simplifying expressions not only helps in solving equations easily but also prepares the polynomial for further operations, like factoring or finding roots, if needed. This systematic approach ensures that expressions are orderly and as concise as possible, which is essential in algebraic problem solving.