Problem 143 Factor completely. \(7 v^{2}-6... [FREE SOLUTION]

Chapter 7: Problem 143

Factor completely. \(7 v^{2}-63 v+56\)

Short Answer

Expert verified

7(v-1)(v-8)

Step by step solution

Identify the Greatest Common Factor (GCF)

First, find the greatest common factor of all the terms in the quadratic expression. The terms are 7, -63, and 56. The GCF of these numbers is 7.

Factor Out the GCF

Factor out the GCF (7) from the expression. This simplifies the expression from: \[7 v^{2}-63 v+56\] to: \[7(v^{2}-9v+8)\]

Factor the Quadratic Expression

Now factor the quadratic expression inside the parentheses. We need to find two numbers that multiply to 8 (the constant term) and add up to -9 (the coefficient of the linear term). These numbers are -1 and -8.

Write the Factored Form

The expression inside the parentheses can be factored as: \[(v-1)(v-8)\]So the completely factored form of the original expression is: \[7(v-1)(v-8)\]

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Key Concepts

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

greatest common factor

The first step in factoring any expression is to find the Greatest Common Factor (GCF). The GCF is the largest number that divides all the terms in the expression without leaving a remainder. In this exercise, we look at the coefficients 7, -63, and 56.
By calculating the GCF, we simplify the expression and make further factoring easier. Here, the GCF of 7, -63, and 56 is 7.

Factoring out the GCF simplifies the quadratic expression from:
7v^{2}-63v+56

to:
7(v^{2}-9v+8).

quadratic expressions

A quadratic expression is a polynomial of the form ax^{2} + bx + c, where a, b, and c are constants. In our simplified expression (v^{2}-9v+8),

a is 1, b is -9, and c is 8.

To factor a quadratic expression, we find two numbers that multiply to the constant term (c) and add up to the coefficient of the linear term (b). In our case, these numbers are -1 and -8, because:

-1 * -8 = 8 (matches c)
-1 + (-8) = -9 (matches b)

So the quadratic expression (v^{2}-9v+8) can be factored into:

(v-1)(v-8).

factoring techniques

Factoring techniques are methods used to break down expressions into their simplest components. This makes solving equations easier. For quadratic expressions, important techniques include:

Finding the Greatest Common Factor (GCF)
Factoring by grouping
Applying special formulas like the difference of squares or perfect square trinomials

In our example, we used the GCF technique first to simplify the initial expression. Then, we factored the quadratic expression inside the parentheses. By combining these techniques, we arrived at the completely factored form:

7(v-1)(v-8).

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91影视

Factor completely. \(7 v^{2}-63 v+56\)

Short Answer

Step by step solution

Identify the Greatest Common Factor (GCF)

Factor Out the GCF

Factor the Quadratic Expression

Write the Factored Form

Key Concepts

greatest common factor

quadratic expressions

factoring techniques

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