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Chapter 7: Problem 79

Factor each trinomial of the form \(x^{2}+b x+c\). \(x^{2}-8 x+7\)

Short Answer

Expert verified
The factorization is \( (x-1)(x-7) \).

Step by step solution

01

Identify coefficients

Identify the coefficients in the trinomial. Here, the trinomial is in the form of \(x^{2}+bx+c\). For the trinomial \(x^{2}-8x+7\), we have \(b=-8\) and \(c=7\).
02

Find factor pairs of c

List the pairs of integers that multiply to \(c=7\). The pairs are \( (1, 7) \) and \( (-1, -7) \).
03

Identify the correct pair

From the factor pairs, identify which pair adds up to \(b=-8\). The pair \((-1, -7)\) adds up to \(-8\).
04

Write the factors

Use the pair \((-1, -7)\) to write the trinomial as a product of two binomials: \((x-1)(x-7)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trinomial Factorization
Trinomial factorization is the process of breaking down a trinomial into the product of two binomials. This means that instead of simply looking at an expression like let's break it down further so it's easier to handle. The given exercise is to factor the trinomial In this case, we identify two numbers that multiply to give the constant term, 7, and add up to give the middle coefficient, We find that These two numbers will then help us split the original trinomial into two binomials: \((x - 1)(x - 7)\).
Factoring correctly will help in solving quadratic equations and other algebraic expressions efficiently.
Algebraic Expressions
Algebraic expressions are a combination of variables, numbers, and arithmetic operations. In the given trinomial factorization problem Here, there are three terms: Each term can include coefficients and variables. Coefficients are numbers multiplied by variables while variables like They can have different coefficients and can be raised to various powers in algebraic expressions. To factorize such expressions, we identify coefficients and constants accurately. This identification lays the foundation for correct factorization further down the line.
Binomial Products
A binomial is an algebraic expression that contains exactly two terms. When we talk about binomial products in the context of trinomial factorization, we mean expressing a trinomial as the product of two binomials. Taking our trinomial we found Thus, it can be factored as the binomial products It's essential to practice the factorization of binomials as it forms the basis for effectively understanding and solving higher-level algebraic problems.
Quadratic Equations
Quadratic equations are polynomial equations of degree 2, typically in the form This equation may often be solved by factorization. In our example, we were working with the quadratic equation by factoring it into binomials we get Setting each binomial to zero gives us the solutions to the equation. This approach is crucial in solving quadratic equations because it often provides the simplest and most intuitive path to find real solutions. Understanding how to factorize and solve these equations will help greatly as you progress further in algebra.

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