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Chapter 7: Problem 64

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

Short Answer

Expert verified
\((y + 1)(y + 7)\)

Step by step solution

01

Identify the coefficients

For the trinomial in the form of \(x^{2}+bx+c\), identify the coefficients. Here, \(y^{2}+8y+7\), \(b\) is 8 and \(c\) is 7.
02

Find two numbers that multiply to \(c\)

Find two numbers that multiply to \(c\) (7) and add up to \(b\) (8). These numbers are 1 and 7 because \(1 \times 7 = 7\) and \(1 + 7 = 8\).
03

Write the factors

Write the trinomial as a product of two binomials using the numbers found. \((y + 1)(y + 7)\).

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Key Concepts

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

Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In algebra, these symbols often represent quantities without fixed values, known as variables.
These variables allow for the formulation of equations and expressions that can describe a wide range of real-world problems and scenarios.
For example, in the trinomial exercise, the variable is 'y'.
Understanding algebraic principles is crucial for solving equations, including quadratic equations.
Quadratic Equations
Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). The solutions to these equations can be found using various methods such as factoring, completing the square, or using the quadratic formula.
In the given exercise, the goal is to factor the quadratic equation \( y^2 + 8y + 7 \). Factoring is one of the simplest and most direct ways to solve quadratics when they can be factored.
Here, the equation \( y^2 + 8y + 7 \) is already set up to be written as a product of two binomials.
Polynomials
Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents of variables.
A common polynomial is the trinomial, which has three terms.
The exercise involves a specific type of polynomial called a quadratic trinomial, \( y^2 + 8y + 7 \).
Understanding the structure of polynomials is key to manipulating and simplifying these expressions, which is essential when factoring.
Factoring Techniques
Factoring is the process of breaking down an equation or expression into simpler components, called factors, that when multiplied together give the original equation.
There are several techniques for factoring polynomials, but for quadratics, a common method is to find two numbers that multiply to the constant term (c) and add up to the linear coefficient (b).
In the exercise, we needed to factor \( y^2 + 8y + 7 \).
The numbers that multiply to 7 (c) and add up to 8 (b) are 1 and 7.
So, the trinomial can be factored into the product of two binomials: \( (y + 1)(y + 7) \).
Practicing these techniques thoroughly can improve your ability to solve more complex equations.

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Most popular questions from this chapter

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