Skip to main content
Ch. P - Fundamental Concepts of Algebra
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. P - Fundamental Concepts of AlgebraProblem 74
Chapter 1, Problem 74

In Exercises 65–92, factor completely, or state that the polynomial is prime. 6x2−6x−12

Verified step by step guidance
1
Factor out the greatest common factor (GCF) from all terms in the polynomial. The GCF of the terms 6x^2, -6x, and -12 is 6. Factor out 6 to get: 6(x^2 - x - 2).
Focus on factoring the quadratic expression inside the parentheses: x^2 - x - 2. Look for two numbers that multiply to -2 (the constant term) and add to -1 (the coefficient of the middle term, x).
The two numbers that satisfy these conditions are -2 and 1. Rewrite the middle term, -x, as -2x + x to split the quadratic expression: x^2 - 2x + x - 2.
Group the terms into two pairs: (x^2 - 2x) and (x - 2). Factor out the greatest common factor from each group. From the first group, factor out x to get x(x - 2). From the second group, factor out 1 to get 1(x - 2).
Combine the factored terms. Since both groups contain the common factor (x - 2), factor it out to get: 6(x - 2)(x + 1). This is the completely factored form of the polynomial.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. Common methods include factoring out the greatest common factor (GCF), using the difference of squares, and applying the quadratic formula for quadratic polynomials.
Recommended video:
07:30
Introduction to Factoring Polynomials

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides all terms of a polynomial. Identifying the GCF is often the first step in factoring, as it simplifies the polynomial and makes it easier to factor further. For example, in the polynomial 6x^2−6x−12, the GCF is 6.
Recommended video:
5:57
Graphs of Common Functions

Quadratic Polynomials

A quadratic polynomial is a polynomial of degree two, typically expressed in the form ax^2 + bx + c. To factor a quadratic, one can look for two numbers that multiply to ac (the product of a and c) and add to b. Understanding the structure of quadratic polynomials is crucial for effective factoring and solving related equations.
Recommended video:
06:36
Solving Quadratic Equations Using The Quadratic Formula
Related Practice
Textbook Question

Write each number in decimal notation without the use of exponents. −3.14×10−3-3.14\(\times\)10^{-3}

Textbook Question

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (−6x3+7x2−9x+3)+(14x3+3x2−11x−7)(-6x^3+7x^2-9x+3)+(14x^3+3x^2-11x-7)

1
views
Textbook Question

Simplify the radical expressions in Exercises 67–74, if possible. ⁵√64x6/⁵√2x

Textbook Question

In Exercises 67–82, find each product. (9x+7y)2

1
views
Textbook Question

Add or subtract terms whenever possible. 425+3254\(\sqrt\)[5]{2} + 3\(\sqrt\)[5]{2}

Textbook Question

State the name of the property illustrated. 6+(−4)=(−4)+6

5
views