Problem 93 Factor each trinomial of the for... [FREE SOLUTION]

Chapter 7: Problem 93

Factor each trinomial of the form \(x^{2}+b x+c\). \(8-6 x+x^{2}\)

Short Answer

Expert verified

The trinomial \( 8 - 6x + x^2 \) factors to \( (x - 2)(x - 4) \).

Step by step solution

- Rewrite the Equation in Standard Form

The given trinomial is not in standard form. The standard form for a quadratic equation is written as: \[ ax^2 + bx + c \]. Rewrite the trinomial to fit this form: \[ 8 - 6x + x^2 \] becomes \[ x^2 - 6x + 8 \].

- Identify the Coefficients

Identify the coefficients of the quadratic equation: \[ a = 1 \], \[ b = -6 \], and \[ c = 8 \].

- Determine Two Numbers that Multiply to c and Add to b

Find two numbers that multiply to the constant term \( c = 8 \) and add up to the coefficient of the linear term \( b = -6 \). The two numbers are \( -2 \) and \( -4 \) because \( -2 * -4 = 8 \) and \( -2 + -4 = -6 \).

- Write the Factored Form

Using the two numbers found in the previous step, write the factored form of the trinomial. \[ x^2 - 6x + 8 \] can be factored as \[ (x - 2)(x - 4) \].

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Key Concepts

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

quadratic equations

Quadratic equations are polynomial equations of the second degree, usually in the format \( ax^2 + bx + c = 0 \). They are recognized by the highest exponent of the variable being 2. Here, the given equation is \( 8 - 6x + x^2 \), which can also be written in the standard form as \( x^2 - 6x + 8 \). The standard form is crucial because it helps in identifying the coefficients needed for various solving methods, such as factoring, using the quadratic formula, or completing the square. In this case, we focus on factoring. Understanding the structure and components of quadratic equations is vital for solving them efficiently.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations. Unlike equations, they do not have an equality sign. In the given exercise, \( 8 - 6x + x^2 \) is an algebraic expression before being equated to zero for solving. When working with algebraic expressions, it's important to recognize like terms and understand how to manipulate them. For example, rearranging \( 8 - 6x + x^2 \) into \( x^2 - 6x + 8 \) helps in the identification of the standard form. The ability to simplify and rewrite algebraic expressions paves the way to easier problem-solving in algebra and beyond.

factoring techniques

Factoring techniques are essential tools for solving polynomial equations. The technique used in the step-by-step solution of the given problem is called factoring trinomials. It involves rewriting a quadratic equation from the standard form \( ax^2 + bx + c \) into a product of two binomials.

First, we identify the coefficients: \( a = 1 \, b = -6 \, \) and \( c = 8 \).
Then, find two numbers that multiply to \( c \) and add up to \( b \). These numbers are \( -2 \) and \( -4 \).
Using these numbers, the equation \( x^2 - 6x + 8 \) factors into \( (x - 2)(x - 4) \).

Mastering factoring techniques allows students to solve not only quadratic equations but also higher-degree polynomials and various algebraic problems. Understanding the reasoning behind each step makes it easier to apply these techniques to different types of equations and expressions.

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91影视

Factor each trinomial of the form \(x^{2}+b x+c\). \(8-6 x+x^{2}\)

Short Answer

Step by step solution

- Rewrite the Equation in Standard Form

- Identify the Coefficients

- Determine Two Numbers that Multiply to c and Add to b

- Write the Factored Form

Key Concepts

quadratic equations

algebraic expressions

factoring techniques

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