Problem 79 - Algebra A Combined Function

Question

Complete each sentence in your own words. If \(x^{2}+b x+c\) is factorable and \(c\) is negative, then the signs of the last-term factors of the binomials are opposite because...

Step-by-Step Solution

Answer

The signs are opposite because a negative product means factors have opposite signs.

Step 1: Understand the Problem

We are given a quadratic expression of the form \(x^{2} + bx + c\) that is factorable. The problem asks about the signs of the factors of the constant term \(c\) in the binomials when \(c\) is negative. The question involves recognizing how the signs of the terms affect factorizability.

Step 2: Recall Factorization Rules

For the quadratic expression \(x^{2} + bx + c\) to be factorable into two binomials, we must find two integers whose product is \(c\) and sum is \(b\). When factoring, the quadratic will take the form \((x + p)(x + q)\), where \(p\times q = c\) and \(p+q = b\).

Step 3: Analyze the Sign of c

Given \(c\) is negative, it implies that the product \(p \times q\) is negative. This can only happen if one of the factors \(p\) or \(q\) is positive and the other is negative, meaning they have opposite signs.

Step 4: Conclude with Opposite Signs

Since the product \(p \times q\) is negative (as specified by the negative \(c\)), the factors of the last term in the binomial must have opposite signs because only opposite signs can result in a negative product.

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial of degree two. It typically looks like this: \(x^2 + bx + c\). Here, "quadratic" refers to anything related to "square," such as the term \(x^2\). Quadratic expressions are fundamental in algebra because they are involved in a wide array of problems, from simple calculations to complex equations in many fields. These expressions are often simplified or manipulated for various solutions. There are specific forms, such as the standard form \(x^2 + bx + c\), where each term has its distinct role:

• \(x^2\): the quadratic term with the variable squared.
• \(bx\): known as the linear term.
• \(c\): the constant term.

Understanding each part of a quadratic expression helps in working out factorizations, calculating roots, and solving equations. Factorizing is particularly useful because it allows you to break down complex expressions into simpler binomials, making calculations easier.

Binomials

In algebra, a binomial is a polynomial with two terms. The word 'binomial' comes from "bi," meaning two, and "nomial," which indicates terms or names.A binomial example might look like \((x + 3)\) or \((x - 2)\). When focusing on quadratic expressions, you often factor these into the form \((x + p)(x + q)\), where each binomial reflects part of the quadratic.

• Each term in the binomial modifies the variable \(x\).
• The process of expanding \((x + p)(x + q)\) helps verify if it equals the original quadratic expression.

By multiplying the terms in the binomials, you recreate the original quadratic expression. Understanding how to work with binomials both in expanding and simplifying expressions is crucial for mastering quadratic functions.

Factorable Quadratics

Factorable quadratics are quadratic expressions that can be expressed as the product of two binomials. A quadratic, \(x^2 + bx + c\), is factorable if you can find two numbers, \(p\) and \(q\), that multiply to \(c\) and add to \(b\).When \(c\) is negative, this means:

• \(p\) and \(q\) must have opposite signs. Only a positive times a negative results in a negative product.
• For example, if \(c = -12\), possible pairs like \( (3, -4), (-3, 4), (2, -6) \) all multiply to \(-12\).
• The right pair also needs to add up to \(b\), balancing the equation.

This process of factorization helps simplify equations, making it easier to find solutions to quadratic problems, discover roots, and ensure every part of your equation is thoroughly understood.