Problem 9 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$x^{2}-5 x-14$$

Step-by-Step Solution

Answer

The polynomial factors as \((x + 2)(x - 7)\).

Step 1: Identify the Polynomial

The given polynomial is \(x^2 - 5x - 14\). It is a quadratic polynomial of the form \(ax^2 + bx + c\) where \(a = 1\), \(b = -5\), and \(c = -14\).

Step 2: Look for Pairs of Factors of the Constant Term

The constant term is \(-14\). We need to find two integers whose product is \(-14\) and whose sum is the coefficient of \(x\), which is \(-5\). Possible pairs of factors of \(-14\) are \((-1, 14), (1, -14), (-2, 7), (2, -7)\).

Step 3: Check Which Pair Sums to the Coefficient of the Linear Term

From the factor pairs in Step 2, the pair (2, -7) sums to -5: \(2 + (-7) = -5\). Therefore, 2 and -7 are the correct factors for the polynomial.

Step 4: Write the Factored Form

Using the integers 2 and -7, we can factor the polynomial as \((x + 2)(x - 7)\). This is because when expanded, \((x + 2)(x - 7) = x^2 - 7x + 2x - 14 = x^2 - 5x - 14\).

Step 5: Verify the Factorization

To ensure accuracy, expand \((x + 2)(x - 7)\) again to check the result: \((x + 2)(x - 7) = x^2 + 2x - 7x - 14 = x^2 - 5x - 14\). This confirms the factorization is correct.

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a type of expression you often encounter in algebra. It has a standard form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The highest power of \(x\) in a quadratic polynomial is 2, which occurs in the term \(ax^2\). This is what gives it the name 'quadratic,' derived from the Latin word for square.

Working with quadratic polynomials involves understanding how the values of the constants affect the graph of the polynomial and its factored form. The process often includes factoring, which means expressing a polynomial as the product of its simpler factors or expressions.

Factoring a quadratic like \(x^2 - 5x - 14\) follows the general strategy of expressing it as two binomials. These will look like \((x + m)(x + n)\), where \(m\) and \(n\) are numbers that you'll determine through factorization.

Integer Factorization

Integer factorization is a critical part of simplifying polynomials, and especially useful when dealing with quadratic ones. The goal of integer factorization is to break down a number into a product of smaller integers, called factors. These factors multiply together to yield the original number.

When factoring a quadratic polynomial, you'll focus on the constant term \(c\) in \(ax^2 + bx + c\). For the quadratic \(x^2 - 5x - 14\), the constant term is \(-14\).

• The task is to find pairs of integers that multiply to \(-14\).
• These numbers must also add up to \(b\), the coefficient of the \(x\) term, which is \(-5\) in this exercise.

By testing different integer pairs, you'll find that (2, -7) work perfectly because their product is \(-14\) and their sum is \(-5\). Identifying such pairs is crucial for successful polynomial factorization.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. A quadratic polynomial is a specific kind of algebraic expression featuring squared terms. Understanding algebraic expressions allows you to manipulate and simplify them through operations like factoring.

When working with expressions such as \(x^2 - 5x - 14\), recognizing patterns and using techniques like factorization transform complex expressions into more manageable forms. Factoring involves writing the polynomial in a different way, using simpler terms.

• An expression like \((x+2)(x-7)\) is the factored form of the polynomial.
• Expanding this expression lets you confirm it's equal to the original quadratic, verifying your work.

Factoring helps in understanding algebraic relationships and finding polynomial roots, valuable skills in both academic settings and real-world problem-solving.