Problem 18 - Algebra 2

Question

Factor each polynomial. \(x^{2}+8 x-9\)

Step-by-Step Solution

Answer

The polynomial \(x^{2}+8x-9\) factors to \((x+9)(x-1)\).

Step 1: Explore the Problem

We are given the quadratic polynomial \(x^{2} + 8x - 9\). The goal is to factor it into two binomials of the form \((x + p)(x + q)\) such that the product of \(p\) and \(q\) equals \(-9\) and their sum equals \(8\).

Step 2: Identify Possible Pairs

We need two numbers whose product is \(-9\) and whose sum is \(8\). Let's list the factors of \(-9\): \((-9, 1)\), \((9, -1)\), \((-3, 3)\), and \((3, -3)\).

Step 3: Check Products and Sums

Calculate the sums for these pairs:- \((-9 + 1 = -8)\)- \((9 + -1 = 8)\)- \((-3 + 3 = 0)\)- \((3 + -3 = 0)\)The pair \((9, -1)\) provides the correct sum of \(8\).

Step 4: Form the Binomials

Using the pair \((9, -1)\), we can write the factors as \((x + 9)(x - 1)\). These two binomials will multiply to give the original polynomial \(x^{2} + 8x - 9\).

Step 5: Verify the Factorization

Verify by expanding \((x + 9)(x - 1)\):\[(x + 9)(x - 1) = x^2 - x + 9x - 9 = x^2 + 8x - 9\]The original polynomial is obtained, which confirms the factorization is correct.

Key Concepts

Quadratic Polynomial

Quadratic polynomials are a special type of polynomial characterized by having a degree of two. This means the highest power of the variable, often denoted as \(x\), is squared. The standard form of a quadratic polynomial is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\).
Quadratics are common in various mathematical contexts, such as algebra and calculus. They are fundamental because they represent parabolic graphs, which have unique features like a vertex and axis of symmetry.
When working with quadratic polynomials, the primary operations involve simplifying, factoring, or solving them for \(x\). In this exercise, the goal is to transform the given quadratic \(x^2 + 8x - 9\) into a product of two simpler expressions.

Binomial Factorization

Binomial factorization involves breaking down a polynomial, like our quadratic, into a product of two simpler binomial expressions. In this context, a binomial is an expression containing exactly two terms. Each factor takes the form \((x + p)(x + q)\), where \(p\) and \(q\) are numbers.
The technique essentially involves reverse engineering the expansion of such binomial products. We aim to find two numbers whose product equals the constant term, \(c\), at the end of the quadratic expression, and whose sum equals the coefficient of \(x\).
In the example exercise, the polynomial \(x^2 + 8x - 9\) was factored using the pair \((9, -1)\). These numbers were selected because their product is \(-9\), and their sum is \(8\), perfectly matching the quadratic's terms. This successful factorization confirms that \((x + 9)(x - 1)\) is indeed the correct result.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition or multiplication. They are the building blocks of algebra, allowing mathematicians to model real-world situations or explore abstract concepts.
Key components in an algebraic expression include terms separated by addition or subtraction symbols. For example, in the expression \(x^2 + 8x - 9\), there are three terms: \(x^2\), \(8x\), and \(-9\). Understanding how to manipulate these terms is central to solving equations and simplifying expressions.
When factoring algebraic expressions like our quadratic, we are engaging in a process that simplifies the equation or prepares it for solving. This often involves identifying common factors, applying formulas, or practicing logical reasoning to express the original formula in a different but equivalent manner.