Factoring

Introduction

Factoring is the process of breaking down a number or an algebraic expression into simpler parts (called factors) that, when multiplied together, give back the original expression. Here are some of the factoring techniques that will be discussed in this lesson.

Factor \bf{25x^2 + 30x}

We begin by finding the greatest common factor (GCF):

Rewrite each term using the GCF (5) (x) = 5x:

Factor out 5x:

Factor \bf{6x^4 - 15x^3 + 3x^2}

We begin by finding the greatest common factor (GCF):

Rewrite each term using GCF (3)(x)(x) = 3x^2:

Factor out 3x² :

Factor \bf { 2l + 2w }

We begin by finding the greatest common factor (GCF):

Rewrite each term using the GCF (2) = 2:

Factor out 2:

Factor \bf {x^2 + 16x + 64}

We begin by identifying the first and last terms as perfect squares:

Recognizing the square of a binomial formula (𝑎 - b)^2 = 𝑎^2 - 2𝑎b + b^2 , we check the middle term:

Since the middle term matches -2(8)(x), we apply the square of binomial formula:

Factor \bf{ 9𝑎^2 b^2 + 24𝑎^2bc + 16c^2}

We begin by identifying the first and last terms as perfect squares:

Recognizing the square of a binomial formula ( 𝑎 + b)² = 𝑎² + 2𝑎b² + b² , we check the middle term:

Since the middle term matches 2(3𝑎^2b) (4c), we apply the square of a binomial formula:

Factor \bf{\dfrac{1}{4}m^2 + \dfrac{1}{5}mn + \dfrac{1}{25}n^2}

We begin by identifying the first

Recognizing the square of a binomial formula ( 𝑎 + b)² = 𝑎² + 2𝑎b + b², we check the middle term:

Factor \bf { x^2 - 25}

We begin by expressing each term as squares:

Recognizing the difference of two squares formula 𝑎² - b² = (𝑎 - b) (𝑎 + b), we rewrite:

Apply the difference of squares formula:

Factor \bf{25x^2 - \dfrac{1}{4}y^2}

We begin by expressing each term as squares:

Recognizing the difference of two square formula 𝑎² - b² = (𝑎 - b)(𝑎 + b), we rewrite:

Apply the difference of square formula:

Factor \bf{9x^2 - 49}

We begin by expressing each term as squares:

Recognizing the difference of two squares formula 𝑎² - b² = (𝑎 - b) (𝑎 + b), we rewrite:

Appy the difference of squares formula:

Factor \bf{x^3 + 64y^3}

We begin by expressing each term as cubes:

Recognizing the sum of cubes formula: 𝑎³ + b³ = (𝑎 + b) (𝑎² - 𝑎b + b²), we identify:

Apply the sum of cubes formula:

Factor \bf{64r^3 - 729s^3}

We begin by expressing each term as cubes:

Recognizing the difference of cubes formula: 𝑎³ - b³ = (𝑎 -b) (𝑎² + 𝑎b +b²), we identify:

Apply the difference of cubes formula:

Factor \bf {125x^6 + 8y^9}

We begin by expressing each term as cubes:

Recognizing the sum of cubes formula: 𝑎³ + b³ = ( 𝑎 + b) ( 𝑎² - 𝑎b + b²), we identify:

Apply the sum of cubes formula:

Factor \bf{16x^4 - 250x}

We begin by finding the greatest common factor (GCF):

The GCF is 2x, so we factor it out:

Next, recognize that 8x³ - 125 is a difference of cubes:

Using the difference of cubes formula:

Apply the formula:

Substituting back, we get:

Final factored form:

Factor \bf{2x^2 - 32}

We begin by finding the greatest common factor (GCF):

The GCF is 2, so we factor it out:

Next, recognize that x² - 16 is a difference of two squares:

Applying this, we get:

Final factored form: