Problem 20 - Algebra A Combined Function

Question

Multiply. $$ (x+7)^{2} $$

Step-by-Step Solution

Answer

\((x+7)^2 = x^2 + 14x + 49\).

Step 1: Identify the Expression to Expand

The expression given is \((x+7)^2\). This is a perfect square trinomial, which can be expanded using the formula \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = x\) and \(b = 7\).

Step 2: Square the First Term

Using the formula \((a+b)^2 = a^2 + 2ab + b^2\), first compute \(a^2\). Since \(a = x\), we have \(a^2 = x^2\).

Step 3: Compute Twice the Product of the Terms

Next, calculate the middle term, which is \(2ab\). Substituting \(a = x\) and \(b = 7\) into \(2ab\), we get: \(2 \times x \times 7 = 14x\).

Step 4: Square the Second Term

Finally, compute \(b^2\). Since \(b = 7\), \(b^2 = 7^2 = 49\).

Step 5: Combine All Terms

Combine the results from Steps 2, 3, and 4: \(x^2 + 14x + 49\). Thus, \((x+7)^2 = x^2 + 14x + 49\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that you get when you square a binomial expression. Let's break it down with the expression

• \((x + 7)^2\)

This expression is a binomial being squared.
The result is always a trinomial, having three terms.
It's called "perfect square" because when you expand it, the terms are predictable and fit the formula

• \((a+b)^2 = a^2 + 2ab + b^2\)

This pattern emerges because we multiply the binomial by itself, which makes it easier to predict the resulting terms.

• The first term \(a^2\) is the square of the first part of the binomial.
• The second term \(2ab\) is twice the product of the two parts.
• The third term \(b^2\) is the square of the last part.

Polynomial Expansion

When you're dealing with polynomial expansion, you're essentially distributing every term of one polynomial by every term of another polynomial.
This is exactly what happens when you expand a perfect square trinomial. Let's look at our exercise's example:

• \((x + 7)^2\)

Expanding this means we'll take each part of \((x+7)\) and multiply it by itself. Break it down as follows:

• First, multiply the first term \(x\) with itself: \(x \times x = x^2\)
• Then, multiply the first term \(x\) with the second: \(x \times 7 = 7x\)
• Now do the same but start with the second term: \(7 \times x = 7x\)
• Finally, multiply the second term by itself: \(7 \times 7 = 49\)

Add these together using their algebraic signs to find:

• \(x^2 + 7x + 7x + 49\)
• Simplify middle terms: \(x^2 + 14x + 49\)

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (such as plus and minus signs). They form the foundation of algebra. Digging into the given expression

• \((x+7)^2\)

it is clear we are working with variables, constants, and operations.
Every polynomial, like a trinomial, is made up of these algebraic expressions.
Understanding the parts of the expression is important:

• **Variables** (in this case, \(x\)) represent unknown numbers that can change or move.
• **Constants** (here 7) are fixed numbers that do not change.
• **Operations** are math processes like addition, subtraction that combine or simplify expressions.

Mastering algebraic expressions means being able to identify the variables, constants, and properly use operations to solve or expand them.
The goal is often to simplify or solve these expressions, leading to their polynomial forms.