Problem 65 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (3 a+1)^{2} $$

Step-by-Step Solution

Answer

The expanded form of \((3a + 1)^2\) is \(9a^2 + 6a + 1\).

Step 1: Understand the Expression

The problem requires us to expand the expression \((3a + 1)^2\). This means taking the binomial \((3a + 1)\) and multiplying it by itself.

Step 2: Apply the Binomial Theorem

Remember that \((x+y)^2 = x^2 + 2xy + y^2\). Here, we can identify \(x = 3a\) and \(y = 1\). The expanded form will be \((3a)^2 + 2(3a)(1) + 1^2\).

Step 3: Calculate Each Term

Now calculate each part of the expanded form: - \((3a)^2 = 9a^2\)- \(2(3a)(1) = 6a\)- \(1^2 = 1\).

Step 4: Formulate the Final Expression

Combine the results from Step 3 to write the final expanded expression: \[ 9a^2 + 6a + 1 \].

Key Concepts

Exponents

Exponents are a fundamental concept in algebra that allow us to express repeated multiplication succinctly. When you see an expression like \((a^n)\), it means that the base \(a\) is multiplied by itself \(n\) times. For example, \(3^2\) is the same as \(3 \times 3\), resulting in 9. In our exercise,

• we dealt with \((3a)^2\), which means \(3a \times 3a\).

Working with exponents involves several basic rules. The most important ones include:

• Product of Powers Rule: When multiplying two powers with the same base, you can add the exponents together. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents, such as \( (a^m)^n = a^{m \times n} \).

These rules help simplify and manage expressions like the one in the exercise. Understanding how to manipulate exponents is key to mastering algebra.

Binomial Theorem

The binomial theorem is a powerful algebraic tool that allows us to expand expressions that involve powers of binomials. A binomial is a two-term expression, like \((x+y)\).When such expressions are raised to a power, the binomial theorem provides a formulaic way to expand them. The formula is expressed as:\[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]Here, \(\binom{n}{k}\) represents the binomial coefficients, and the sum runs from \(k=0\) to \(n\). In our example,

• we had \((3a + 1)^2\), which is an application of a simple special case of the binomial theorem, \((x+y)^2\).

The expanded form from our exercise was obtained using:

• \(x^2 + 2xy + y^2\), where \(x = 3a\) and \(y = 1\).

Understanding the binomial theorem can greatly simplify the process of expanding polynomials, especially for higher powers.

Algebraic Expressions

Algebraic expressions are like the building blocks of algebra. They are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. An expression such as \((3a + 1)\) is considered a binomial because it contains two terms.In solving algebraic expressions, it is crucial to understand the structure:

• Terms: Pieces of the expression separated by plus or minus signs. In our example, \(3a\) and \(1\) are terms.
• Coefficients: Numbers in front of the variables (like 3 in \(3a\)).
• Variables: Symbols that represent numbers and can vary in value. In \(3a + 1\), \(a\) is the variable.

The ultimate goal is to manipulate these expressions to simplify, expand, or solve them. Understanding how to combine like terms and apply the distributive property makes working with algebraic expressions more manageable.