Problem 110 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ (x+3)^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified expression for \((x+3)^2\) is \(x^2 + 6x + 9\).

Step 1: Identify the binomial terms

Here, we have a binomial expression of the form \((x+3)^2\). Our \(a\) term corresponds to \(x\) and our \(b\) term corresponds to \(3\), and \(n=2\).

Step 2: Apply the binomial expansion formula

We will now use the binomial expansion formula for \(n=2\): \((a+b)^2 = \binom{2}{0}a^2 b^0 + \binom{2}{1}a^1 b^1 + \binom{2}{2}a^0 b^2\). Substitute \(a=x\) and \(b=3\) into the formula to obtain: $$ (x+3)^2 = \binom{2}{0}x^2(3)^0 + \binom{2}{1}x^1(3)^1 + \binom{2}{2}x^0(3)^2 $$

Step 3: Evaluate the binomial coefficients

Calculate the binomial coefficients and coefficients for each term: \(\binom{2}{0} = 1\), \(\binom{2}{1} = 2\), \(\binom{2}{2} = 1\). The expression is: $$ (x+3)^2 = 1x^2(3)^0 + 2x^1(3)^1 + 1x^0(3)^2 $$

Step 4: Simplify the expression

Simplify the expression by calculating the powers and multiplying the coefficients: $$ (x+3)^2 = 1x^2(1) + 2x(3) + 1(9) $$ $$ (x+3)^2 = x^2 + 6x + 9 $$ The simplified expression for \((x+3)^2\) is \(x^2 + 6x + 9\).

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra. It allows us to expand expressions that are raised to a power. For example, the expression \((x+3)^2\) is a binomial, which means it has two terms: \(x\) and \(3\). By using the binomial theorem, we can expand this expression without directly multiplying it out.

The theorem provides a formula: \[(a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1} b^1 + \cdots + \binom{n}{n}a^0 b^n\] where:

• \(a\) and \(b\) are terms in the binomial.
• \(n\) is the power the binomial is raised to.
• \(\binom{n}{k}\) is a binomial coefficient.

This formula helps simplify calculations significantly, especially for higher powers. It reduces the need to perform long multiplications for each term.

Like Terms

In algebra, 'like terms' refer to terms in an expression that have the same variables raised to the same powers. For instance, in the expression \(x^2 + 6x + 9\), \(x^2\) is different from \(6x\) due to different exponents and terms. However, if there were multiple terms like \(6x + 3x\), they are like terms since they both contain \(x\) raised to the same power (here, the power of 1).

Combining like terms simplifies algebraic expressions. You simply add or subtract the coefficients, which are the numerical parts, of these like terms.

• Example: Combining \(6x\) and \(3x\) gives you \(9x\).
• Remember: Only terms with exactly the same variables and powers can be combined.

Simplifying expressions by combining like terms makes them easier to work with in further calculations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. For example, \(x^2 + 6x + 9\) is an algebraic expression.

Key components of algebraic expressions include:

• **Variables (like \(x\), \(y\))**: Represent unknowns or values that can change.
• **Constants (like \(3\), \(9\))**: These are fixed values that do not change.
• **Coefficients (like \(6\) in \(6x\))**: These numbers multiply the variables.
• **Operations (like addition or subtraction)**: Combine the numbers and variables.

Simplifying algebraic expressions involves reducing them to the simplest form. This is done by combining like terms and applying operations where possible. Understanding how to construct and simplify these expressions is foundational in algebra, as it sets the basis for more complex operations and problem-solving techniques.