6 Questions
What is the purpose of finding the Greatest Common Factor (GCF) of a polynomial?
To simplify the polynomial and make it easier to work with
What is the factored form of the quadratic expression x^2 + 5x + 6?
(x + 3)(x + 2)
What is the formula for factoring a difference of cubes?
(a - b)(a^2 + ab + b^2)
How can the GCF be used to simplify a polynomial?
By dividing each term by the GCF
What is the factored form of the sum of cubes x^3 + 8?
(x + 2)(x^2 - 2x + 4)
What is the purpose of factoring quadratic expressions?
To simplify the quadratic expression and make it easier to work with
Study Notes
Greatest Common Factor (GCF)
- The GCF is the largest number that divides two or more integers without leaving a remainder.
- To find the GCF of a polynomial, list the factors of each term and identify the common factors.
- The GCF can be a single variable, a constant, or a combination of variables and constants.
Factoring Out Greatest Common Factor
- Factoring out the GCF involves dividing each term of the polynomial by the GCF.
- This is useful for simplifying polynomials and making them easier to work with.
- Example: Factor out the GCF from 6x^2 + 12x
- GCF = 6x
- Factored form: 6x(x + 2)
Factoring Quadratic Expressions
- A quadratic expression is a polynomial of degree two, in the form ax^2 + bx + c.
- Factoring quadratic expressions involves expressing them as a product of two binomials.
- There are several methods for factoring quadratic expressions, including:
- Factoring out the GCF
- Using the "ac method" (where a and c are the coefficients of the x^2 and constant terms)
- Using the " guessing method" (where you try different combinations of factors)
- Example: Factor the quadratic expression x^2 + 5x + 6
- Factored form: (x + 3)(x + 2)
Factoring Sums and Differences of Cubes
- A sum of cubes is a polynomial in the form a^3 + b^3.
- A difference of cubes is a polynomial in the form a^3 - b^3.
- Factoring sums and differences of cubes involves using the following formulas:
- a^3 + b^3 = (a + b)(a^2 - ab + b^2)
- a^3 - b^3 = (a - b)(a^2 + ab + b^2)
- Example: Factor the sum of cubes x^3 + 8
- Factored form: (x + 2)(x^2 - 2x + 4)
Greatest Common Factor (GCF)
- The GCF is the largest number that divides two or more integers without leaving a remainder.
- To find the GCF of a polynomial, list the factors of each term and identify the common factors.
- The GCF can be a single variable, a constant, or a combination of variables and constants.
Factoring Out Greatest Common Factor
- Factoring out the GCF involves dividing each term of the polynomial by the GCF.
- This process simplifies polynomials and makes them easier to work with.
- Example: Factoring out the GCF from 6x^2 + 12x results in 6x(x + 2).
Factoring Quadratic Expressions
- A quadratic expression is a polynomial of degree two, in the form ax^2 + bx + c.
- Factoring quadratic expressions involves expressing them as a product of two binomials.
- Methods for factoring quadratic expressions include:
- Factoring out the GCF.
- Using the "ac method" (where a and c are the coefficients of the x^2 and constant terms).
- Using the "guessing method" (where you try different combinations of factors).
- Example: Factoring the quadratic expression x^2 + 5x + 6 results in (x + 3)(x + 2).
Factoring Sums and Differences of Cubes
- A sum of cubes is a polynomial in the form a^3 + b^3.
- A difference of cubes is a polynomial in the form a^3 - b^3.
- Factoring sums and differences of cubes involves using the formulas:
- a^3 + b^3 = (a + b)(a^2 - ab + b^2).
- a^3 - b^3 = (a - b)(a^2 + ab + b^2).
- Example: Factoring the sum of cubes x^3 + 8 results in (x + 2)(x^2 - 2x + 4).
Learn about finding the Greatest Common Factor (GCF) of polynomials and how to factor it out to simplify expressions.
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