Problem 130 - Algebra and Trigonometry Real Mathematics, Real People

Question

Factor the polynomial. $$x^{2}+4 x-5$$

Step-by-Step Solution

Answer

The factorized form of the given polynomial \( x^{2}+4x-5 \) is \( (x-1)(x+5) \)

Step 1: Identifying the Constants

In this polynomial, \( x^{2}+4x-5 \), the constants 'a', 'b' and 'c' are 1, 4 and -5 respectively.

Step 2: Finding Factors

The next step involves identifying the two numbers, which when multiplied equal to -5 (product of 'a' and 'c') and when added equal to 4. The numbers satisfying these conditions are 5 and -1, because \(5 \times -1 = -5\), and \(5 + (-1) = 4\).

Step 3: Writing the Polynomial in Factorized Form

The identified numbers are the coefficients for the factored form of the polynomial. So the factorized form of the polynomial \( x^{2}+4x-5 \) will be \( (x-1)(x+5) \)

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a specific type of polynomial equation. They take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
This means that the equation includes a squared term (\(x^2\)) as its highest power.
Quadratic equations play a crucial role in algebra. They appear often in real-world problems. Solving these equations involves finding values of \(x\) that make the equation true. These values are known as the "roots" or "solutions" of the equation.
There are several methods to solve quadratic equations. Factoring, completing the square, and using the quadratic formula are popular options. Understanding the structure can help you choose the best method for solutions.

Exploring Factoring Techniques

Factoring is one of the key techniques used to solve quadratic equations. The goal is to write the quadratic equation as a product of two simpler expressions. This process makes finding the roots straightforward.
In the polynomial \(x^2 + 4x - 5\), the factors \((x-1)(x+5)\) represent the solution process. Here's how factoring usually works for quadratics:

• Identify two numbers that multiply to \(a \times c\) and add to \(b\).
• Rewrite the middle term using these numbers.
• Factor by grouping the terms into pairs.

Factoring is effective when the quadratic can be easily expressed as the product of two expressions. However, it may not always be straightforward, especially with more complex equations.

The Role of Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebra and appear in many mathematical problems.
In a quadratic equation, expressions like \(x^2 + 4x - 5\) include:

• Variables (\(x\))
• Coefficients (numbers that multiply the variables, like 1 and 4)
• Constants (numbers without variables, like -5)

Understanding algebraic expressions allows you to manipulate and simplify them. These skills are crucial for solving equations, performing calculations, and expressing complex ideas clearly.
Mastering algebraic expressions facilitates solving equations like quadratics and determining their factored forms, as seen in the exercise.