Questions and Answers
What is the first step when factoring any polynomial?
Factor out the GCF
The GCF of the polynomial 2x^3 - 7x^2 + 3x is ___
x
The GCF of the polynomial 33x^4 - 22 is ___
11
The GCF of the polynomial 24x^5 - 56x^3 + 16x is ___
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Which polynomials are prime? (Check all that apply)
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Which is the completely factored polynomial for 4x^3 - 12x^2 + 7x - 21?
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Which diagram shows the X method being used correctly to factor the trinomial 2x^2 - 3x + 1?
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What is the GCF of the polynomial 6x^2 - 8x + 2?
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What is the equivalent factored form of the polynomial 6x^2 - 8x + 2?
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Completely factor the polynomial 8x^2 - 24x + 20x - 60.
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Which product of prime polynomials is equivalent to 30x^3 - 5x^2 - 60x?
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Which are the prime factors of the polynomial 24x^4 - 3x? (Check all that apply)
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Has Leo factored the polynomial 3x^3 - 3x + 5x^2 - 5 completely?
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Study Notes
Factoring Polynomials Overview
- The first step in factoring any polynomial is to identify and factor out the GCF (Greatest Common Factor).
- Factoring out the GCF simplifies the polynomial, making further factorization easier.
Finding the GCF
- For the polynomial 2x³ - 7x² + 3x, the GCF is x.
- For 33x⁴ - 22, the GCF is 11.
- For 24x⁵ - 56x³ + 16x, the GCF is 8x.
Identifying Prime Polynomials
- The following polynomials are considered prime:
- 2x² + 7x + 1
- 3x² + 8
- x² + 36
Complete Factorization
- The polynomial 4x³ - 12x² + 7x - 21 can be factored as (4x² + 7)(x - 3).
- The completely factored form of this polynomial is choice D.
Using the X Method
- The correct diagram for applying the X method to factor the trinomial 2x² - 3x + 1 is represented by C.
GCF and Factored Form of a Polynomial
- For the polynomial 6x² - 8x + 2, the GCF is 2.
- The equivalent factored form is 2(3x - 1)(x - 1).
Factoring a Four-Term Polynomial
- The polynomial 8x² - 24x + 20x - 60 completely factors to 4(x - 3)(2x + 5) (choice D).
Equivalent Products of Prime Polynomials
- The product of prime polynomials equivalent to 30x³ - 5x² - 60x is 5x(2x - 3)(3x + 4) (choice C).
Identifying Prime Factors
- The prime factors of the polynomial 24x⁴ - 3x include:
- 3x
- (2x - 1)
- (4x² + 2x + 1)
Complete Factorization Check
- Leo's factored form of 3x³ - 3x + 5x² - 5 using double grouping is not completely factored.
- The term x² - 1 can be further factored into (x - 1)(x + 1) leading to (x - 1)(x + 1)(3x + 5) as the complete factorization.
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Description
This quiz covers the basics of factoring polynomials, including identifying the greatest common factor (GCF) and recognizing prime polynomials. It also delves into complete factorization techniques and methods such as the X method. Test your understanding of these fundamental concepts in polynomial algebra.
