Problem 86 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$(z+9)^{2}$$

Step-by-Step Solution

Answer

\((z+9)^2 = z^2 + 18z + 81\).

Step 1: Write the Expression

The given expression is \[(z+9)^{2}\]This is in the form of \[(a+b)^2\]where \(a = z\) and \(b = 9\).

Step 2: Use the Binomial Formula

Expand the expression using the binomial formula \[(a+b)^2 = a^2 + 2ab + b^2\]Let \(a = z\) and \(b = 9\). Substitute the values to get:\[z^2 + 2(z)(9) + 9^2\]

Step 3: Calculate Each Term

Let's calculate each term separately:1. \(z^2 = z^2\)2. \(2 \times z \times 9 = 18z\)3. \(9^2 = 81\)

Step 4: Combine All Terms

Combine the calculated terms to get:\[z^2 + 18z + 81\]Thus, \[(z+9)^2 = z^2 + 18z + 81\]

Key Concepts

Polynomial Expansion

Polynomial expansion is a fundamental concept in algebra that helps in breaking down expressions into simpler terms. In the context of the binomial theorem, polynomial expansion allows us to expand expressions like \[(a + b)^n\] into a sum of terms. In our exercise, we worked with \[(z+9)^2\], which is a binomial consisting of two terms. This expression indicates that we will multiply \((z+9)\) by itself.

• Using the binomial formula, we can simplify this process and avoid lengthy multiplication.
• The formula \[(a + b)^2 = a^2 + 2ab + b^2\] allows us to expand the expression in one step.
• Each term is derived by using powers and products of the original terms \(a\) and \(b\).

Understanding polynomial expansion can significantly simplify solving algebraic expressions, particularly as they become more complex in higher dimensions or higher powers.

Exponents

Exponents are a shorthand way to express repeated multiplication. By learning how to work with them, computations become easier and faster. An exponent indicates how many times a number (the base) is multiplied by itself. For instance, in our exercise,\[(z+9)^2\], the exponent \(2\) tells us to multiply \((z+9)\) by itself.

• Understanding that \((z+9)^2\) involves squaring the base is crucial for accurate computations.
• The binomial formula then helps manage the multiplication of separate terms within the base expression.
• Exponents often follow precise rules — for instance, any base to the power of zero equals one, and multiplying same bases means adding their exponents.

Thus, mastering the basics of exponents equips students with the skills needed to tackle more complex algebraic problems efficiently.

Algebraic Expression

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or set of values. In this exercise, the expression \((z + 9)^2\) is an algebraic expression involving a variable \(z\) and a constant \(9\).

• An algebraic expression must be manipulated correctly using arithmetic operations and mathematical rules to solve or simplify them.
• The term \(z^2\) in the expansion means that \(z\) is squared, representing a product of the variable by itself.
• Collective terms like \(2(z)(9)\) are formed from multiplying constants with variables.

Algebraic expressions enable us to describe various quantities mathematically and find unknown values by manipulating these expressions according to algebra's structured rules.