Problem 46 - Algebra 1: Concepts and Skills

Question

Factor \(x^{2}-10 x-24\) $$ a.\quad(x-4)(x-6) $$ $$ b.\quad(x+4)(x+6) $$ $$ c.\quad(x+2)(x-12) $$ $$ d.\quad(x-2)(x+12) $$

Step-by-Step Solution

Answer

The correct answer is \(x^{2}-10x-24 = (x - 4)(x - 6)\), so option (a) is correct.

Step 1: Identify the Coefficients

Identify coefficients \(a\), \(b\), and \(c\) in the polynomial \(x^{2}-10x-24\). Here, \(a = 1\), \(b = -10\), and \(c = -24\).

Step 2: Find Factor Pairs of Coefficient c

Find pairs of factors that multiply to get coefficient \(c\), which is \(-24\). The factor pairs of \(-24\) are \((-1, 24), (1, -24), (-2, 12), (2, -12), (-3, 8), (3, -8), (-4, 6), and (4, -6)\).

Step 3: Choose the Correct Pair of Factors

Choose the pair of factors that adds up to coefficient \(b\), which is \(-10\). The pair that matches this is \(-4\) and \(-6\).

Step 4: Write the Polynomial as a Product of Two Binomials

Write the polynomial as a product of two binomials, \(x^{2}-10x-24 = (x - 4)(x - 6)\).

Key Concepts

Polynomials

Polynomials are expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication. In our example, we have the polynomial \(x^2 - 10x - 24\). This specific case is a quadratic polynomial because the highest exponent of the variable \(x\) is 2. Quadratic polynomials have the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients.
Typically, these coefficients are real numbers, and for most basic exercises, they are integers. The goal in factoring polynomials is to express them as products of simpler expressions. This process helps in solving equations, simplifying expressions, and understanding polynomial graphs.
Understanding what a polynomial is and how it is structured is fundamental in tackling problems like factoring, graphing, and solving polynomial equations. Recognizing the degree and the coefficients plays a crucial role in selecting the right method for solving or simplifying these expressions.

Factor Pairs

Factor pairs are two numbers, which when multiplied together, give a specific product. In the context of polynomials, finding factor pairs is crucial for factorization. For the polynomial \( x^2 - 10x - 24 \), we focus on the constant term, \(-24\), which we need to factor.
Identifying factor pairs involves listing all possible combinations of numbers that multiply to \(-24\). These are:

• \((-1, 24)\) and \((1, -24)\)
• \((-2, 12)\) and \((2, -12)\)
• \((-3, 8)\) and \((3, -8)\)
• \((-4, 6)\) and \((4, -6)\)

From these pairs, we look for the ones that add up to the linear coefficient in the polynomial, \(-10\). Only the pair \((-4, -6)\) satisfies this condition, as their sum is indeed \(-10\). This is a key step in the process of factoring quadratics, as finding the correct pair of factors is essential for breaking down the quadratic into simpler binomials.

Quadratic Expressions

Quadratic expressions are a type of polynomial characterized by their degree of 2, meaning they will include an \(x^2\) term. Factoring quadratic expressions like \(x^2 - 10x - 24\) involves rewriting this expression as a product of two binomial expressions. This can simplify solving quadratic equations and analyzing the polynomial's behavior.
To factor a quadratic expression correctly, we make use of factor pairs, as mentioned earlier, and the method of finding binomials that when multiplied together recreate the original expression. Here, we have determined that \(x^2 - 10x - 24\) factors into \((x - 4)(x - 6)\).
Essentially, factoring translates a complex quadratic into simpler parts, easing up the task of finding roots and sketching graphs. Recognizing these factored forms allows one to solve equations like \((x - 4)(x - 6) = 0\) quickly, finding that \(x = 4\) or \(x = 6\) are solutions. This understanding aids in interceptive graphing and in evaluating the polynomial's behavior across different domains and ranges.