Problem 35 - Algebra 2

Question

Simplify each expression. $$ (x-7)(x+9) $$

Step-by-Step Solution

Answer

The simplified expression is \(x^2 + 2x - 63\).

Step 1: Apply the Distributive Property

We start by using the distributive property, which says that \((a + b)(c + d) = ac + ad + bc + bd\).For the expression \((x - 7)(x + 9)\), distribute each term in the first parenthesis across each term in the second parenthesis:1. Multiply the first terms: \(x \cdot x = x^2\).2. Multiply the outer terms: \(x \cdot 9 = 9x\).3. Multiply the inner terms: \(-7 \cdot x = -7x\).4. Multiply the last terms: \(-7 \cdot 9 = -63\). Doing this gives us: \(x^2 + 9x - 7x - 63\).

Step 2: Combine Like Terms

Now, look at the expression obtained after distribution: \(x^2 + 9x - 7x - 63\).You can combine the like terms, which are the terms with the variable \(x\).Combine \(9x - 7x\) which simplifies to \(2x\).Thus, the expression simplifies to \(x^2 + 2x - 63\).

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that allows us to simplify expressions with ease. Imagine it as breaking down a complex multiplication problem into smaller, more manageable parts. The property states that for any numbers or variables, the expression \(a(b + c)\) is equivalent to \(ab + ac\).

This property is extremely useful when dealing with polynomials, which are algebraic expressions made up of terms. In our case, when simplifying \((x - 7)(x + 9)\), the distributive property helps to expand this expression into individual terms: \(x \cdot x\), \(x \cdot 9\), \(-7 \cdot x\), and \(-7 \cdot 9\).

Think of each term as a piece of a puzzle. By multiplying each component and adding them together, we can fully express the original polynomial in its expanded form.

Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power. In the expression \(x^2 + 9x - 7x - 63\), we look for terms with the same variable component.

Here, \(+9x\) and \(-7x\) are like terms because they both contain the same variable \(x\) raised to the first power. To combine them, it's like adding and subtracting normal numbers:

• Add the coefficients of identical variable terms together. \(9x - 7x\) simplifies to \(2x\).

By combining like terms, the expression becomes even simpler: \(x^2 + 2x - 63\). This process helps reduce the complexity of the expression, making it easier to understand and work with further.

Factoring

Factoring is the process of breaking down a polynomial into simpler terms, often called factors, that when multiplied together, produce the original polynomial. It can be thought of as the reverse process of distributing. In our case, while the problem requires expanding the expression \((x - 7)(x + 9)\), understanding factoring can help contextualize the simplified form we arrived at.

Once the expression is simplified to \(x^2 + 2x - 63\), factoring can actually help in breaking it back down into binomials if needed, or used for solving equations. A factored expression is easier to solve, especially when set equal to zero. This is because each factor represents a potential solution.

Though our exercise didn't require factoring back the expression, recognizing the layout of a factored expression, like our starting point \((x-7)(x+9)\), gives a full circle understanding of polynomial expressions in algebra.