Problem 18 - Elementary Algebra

Question

Are the following factored correctly? Check by multiplying. $$ x_{2}+3 x-10=(x+5)(x-2) $$

Step-by-Step Solution

Answer

The factorization is correct.

Step 1: Expand the Given Factors

To verify the factorization, we'll start by expanding the right-hand side: \ \ The given factors are \((x+5)(x-2)\). To multiply these, we use the distributive property: \[(x + 5)(x - 2) = x \cdot x + x \cdot (-2) + 5 \cdot x + 5 \cdot (-2)\] Compute each term: - First term: \(x \cdot x = x^2\)" - Second term: \(x \cdot (-2) = -2x\) - Third term: \(5 \cdot x = 5x\) - Fourth term: \(5 \cdot (-2) = -10\)

Step 2: Simplify the Expression

Now, combine the like terms from the expansion: \[x^2 + (-2x) + 5x - 10\] Simplify by adding the like terms: - Combine \(-2x\) and \(5x\) to get \(3x\) Thus, the expression simplifies to: \[x^2 + 3x - 10\]

Step 3: Compare Both Sides

Compare the expanded expression \(x^2 + 3x - 10\) with the original polynomial \(x^2 + 3x - 10\). Both expressions are identical.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by a set of terms inside a parenthesis. This property states that you multiply the outside term with each term inside the parentheses individually, then sum up the results.
When checking if a polynomial has been factored correctly, we use the distributive property to expand the terms. In the given expression

• the polynomial is supposedly factored as \((x + 5)(x - 2)\), and we need to verify this by expanding it.

Applying the distributive property here means distributing each term in one bracket to every term in the other bracket. This results in the multiplication of every term combination possible. This is a crucial technique for verifying or simplifying algebraic expressions. Understanding how to apply this property ensures accuracy when working with various polynomial operations.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients. These expressions can have terms made up of variables, constants, or a combination of both, raised to whole number exponents. In essence, a polynomial is a sum of several terms.
For instance, the polynomial in the problem is \(x^2 + 3x - 10\), which is a quadratic polynomial due to the highest exponent being 2.
Polynomials can vary in complexity:

• They can be as simple as \(x + 1\) or as complex as \(4x^3 + 3x^2 - 5x + 7\).

Understanding the structure and behavior of polynomials is key to factoring them effectively. The goal of factoring is to express a polynomial as a product of simpler polynomials, providing insight into its roots and properties. Factoring polynomials involves tactics such as identifying common factors, using the distributive property, or employing special formulas, hence gaining mastery over them improves your ability to solve various algebraic equations.

Multiplying Binomials

Multiplying binomials is a specific application of the distributive property where both expressions consist of two terms each. The process involves multiplying each term in the first binomial by each term in the second binomial.
A common method used here is known as FOIL, which stands for First, Outer, Inner, Last. This technique helps students remember which terms to multiply:

• **First**: Multiplying the first terms of each binomial.
• **Outer**: Multiplying the outer terms.
• **Inner**: Multiplying the inner terms.
• **Last**: Multiplying the last terms of each binomial.

So for the expression \((x + 5)(x - 2)\), using the FOIL method translates into:- First: \(x \cdot x = x^2\)- Outer: \(x \cdot (-2) = -2x\)- Inner: \(5 \cdot x = 5x\)- Last: \(5 \cdot (-2) = -10\) Adding these results gives: \(x^2 - 2x + 5x - 10\), which simplifies to the polynomial \(x^2 + 3x - 10\). Multiplying binomials correctly ensures that you factor and expand expressions accurately in algebraic equations.