Algebra 1 - Factoring Polynomials Flashcards

Questions and Answers

What are the two methods to find the prime factorization of a number?

Factor tree, Ladder diagram

Find the prime factorization of 40.

2³ * 5

Find the prime factorization of 17.

What are the two methods to find the GCF of a pair of numbers?

List the factors, Prime Factorization Signup and view all the answers

Find the GCF of 12 and 16.

4 Signup and view all the answers

How do you find the GCF of a pair of monomials?

Align the common factors and find their product. Signup and view all the answers

Find the GCF of 6x² and 7y³.

1 Signup and view all the answers

Find the GCF of 18x² and 27x³.

9x² Signup and view all the answers

What property states that a(b+c) = ab + ac?

The Distributive Property Signup and view all the answers

What happens to the distributive property when you factor by GCF?

It is reversed. Signup and view all the answers

What are the steps to factoring using the GCF?

Find the GCF, write terms as products, use the distributive property, check answer. Signup and view all the answers

Factor 8x³ - 4x² - 16x by GCF.

4x(2x² - x - 4) Signup and view all the answers

What are the steps to factoring by grouping?

Group terms with common factors, factor out GCF, factor the common binomial, check answer. Signup and view all the answers

Factor 6x⁴ - 4x³ + 12x - 8 by grouping.

(3x - 2)(2x³ + 4) Signup and view all the answers

What happens when you factor two quantities that are opposites?

Their sum is 0. Signup and view all the answers

Factor 15x² - 10x³ + 8x - 12.

(3 - 2x)(5x² - 4) Signup and view all the answers

What is the form to factor a trinomial?

x² + bx + c Signup and view all the answers

Factor x² - 4x - 32.

(x - 8)(x + 4) Signup and view all the answers

Factor 20x² + 21x - 5.

(4x + 5)(5x - 1) Signup and view all the answers

What is a polynomial?

A term or the sum or difference of terms Signup and view all the answers

What is a monomial?

An expression with one term Signup and view all the answers

What is a binomial?

A polynomial with two terms Signup and view all the answers

What is a trinomial?

A polynomial with three terms Signup and view all the answers

What are factors?

Numbers or variables being multiplied Signup and view all the answers

What does it mean to factor?

To write an expression as the product of its factors Signup and view all the answers

What is the Greatest Common Factor (GCF)?

The largest common factor of two or more terms Signup and view all the answers

What is the Zero Product Property?

If the product of two numbers is zero, then one of the numbers must be zero Signup and view all the answers

Study Notes

Prime Factorization

Two methods: Factor tree and Ladder diagram.
Example: Prime factorization of 40 is (2^3 \times 5) or (5 \times 2 \times 2 \times 2).
Prime factorization of 17 is simply 17 or (17 \times 1).

Greatest Common Factor (GCF)

Two methods to find GCF: List the factors or use Prime Factorization.
GCF of 12 and 16 is 4, derived from their factor lists.
For monomials, find the GCF by:
- Writing prime factorizations of coefficients.
- Aligning common factors.
- Multiplying the common factors.

GCF Examples

GCF of (6x^2) and (7y^3) is 1 (no common factors).
GCF of (18x^2) and (27x^3) is (9x^2).

Properties

Distributive Property: (a(b+c) = ab + ac).
Factoring by GCF reverses the distributive property: (xy + xz = x(y+z)).

Factoring Techniques

Steps for factoring using GCF:
- Find the GCF of the polynomial terms.
- Write terms as products using the GCF.
- Apply the Distributive Property to factor out the GCF.
- Verify the answer.
Example of factoring (8x^3 - 4x^2 - 16x) yields (4x(2x^2 - x - 4)).

Factoring by Grouping

Steps for factoring by grouping:
- Group terms with common factors.
- Factor out the GCF from each group.
- Factor out the common binomial factor.
- Check solution.
Example: Factoring (6x^4 - 4x^3 + 12x - 8) results in ((3x - 2)(2x^3 + 4)).

Special Cases

When factoring two opposing quantities, the sum is 0.
Example: Factoring (15x^2 - 10x^3 + 8x - 12) results in ((3 - 2x)(5x^2 - 4)).

Trinomials and Polynomial Types

Standard form for factoring a trinomial: (x^2 + bx + c).
Example: Factoring (x^2 - 4x - 32) gives ((x - 8)(x + 4)).
Example: Factoring (20x^2 + 21x - 5) results in ((4x + 5)(5x - 1)).

Definitions

Polynomial: Combination of terms through addition or subtraction.
Monomial: One-term expression (number, variable, or product).
Binomial: Polynomial consisting of two terms.
Trinomial: Polynomial consisting of three terms.
Factors: Elements being multiplied in an expression.
GCF: Largest common factor among terms.
Zero Product Property: If a product equals zero, at least one factor must be zero.

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Description

Test your understanding of factoring polynomials with these flashcards. This quiz includes methods like factor trees and ladder diagrams, as well as finding GCFs and prime factorizations. Perfect for Algebra 1 students looking to reinforce their knowledge.