##### Laura Amatulli

(Rochester Hills - United States)Teaching 8th grade in a middle school in suburban Detroit I have strong interests in Earth Science and leadership. I have been a teacher consultant for our local National Writing Project site, Meadow Brook Writing Project, which keeps me active ...

## Factoring binomials and polynomials

**Description:**

^{rd}degree polynomial.

**Last Updated:**

**Subject(s):**- Mathematics
- Mathematics > Algebra

**Educational Level(s):**- Grades 9-10 / Ages 14-16
- Grades 11-12 / Ages 16-18

**Instructional Component Type(s):**- Curriculum: Unit

- Contributed By: Winpossible

### Getting started

**Description:**

In this mini-lesson you'll learn how to factor the binomials and polynomials. It is an important step in solving problems in a good number of algebraic applications. In factoring polynomials, we determine all the terms that were multiplied together to get the given polynomial. We then try to factor each of the terms we found in the first step. This continues until we can’t factor anymore. For Example, here is the complete factorization of the polynomial

x^{4} – 16 = (x^{2} + 4)(x + 2)(x – 2)

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Algebra I. Click on the video below to go through it. If you like it, you can buy our online course in Algebra I by clicking here.

**Last Updated:**

**Subject(s):**- Mathematics
- Mathematics > Algebra

**Educational Level(s):**- Grades 9-10 / Ages 14-16
- Grades 11-12 / Ages 16-18

**Instructional Component Type(s):**- Curriculum: Lesson Plan

In this mini-lesson you'll learn how to factor the binomials and polynomials. It is an important step in solving problems in a good number of algebraic applications. In factoring polynomials, we determine all the terms that were multiplied together to get the given polynomial. We then try to factor each of the terms we found in the first step. This continues until we can’t factor anymore. For Example, here is the complete factorization of the polynomial

x^{4} – 16 = (x^{2} + 4)(x + 2)(x – 2)

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Algebra I. Click on the video below to go through it. If you like it, you can buy our online course in Algebra I by clicking here.

### Factoring an expression of exponents with the same base

**Description:**

In this mini-lesson you'll learn how to factor an expression of exponents with the same base. Generally speaking, for factoring exponents with the same base, we need to take a common factor out of the expression, and this factor is the base raised to the lowest exponent. For example, x

^{2}is the common factor in the expression x

^{2}+ x

^{6}, and hence the expression can be rewritten as x

^{2}(1 + x

^{4}).

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Algebra I. Click on the video below to go through it. If you like it, you can buy our online course in Algebra I by clicking here.

**Last Updated:**

**Subject(s):**- Mathematics
- Mathematics > Algebra

**Educational Level(s):**- Grades 9-10 / Ages 14-16
- Grades 11-12 / Ages 16-18

**Instructional Component Type(s):**- Curriculum: Lesson Plan

In this mini-lesson you'll learn how to factor an expression of exponents with the same base. Generally speaking, for factoring exponents with the same base, we need to take a common factor out of the expression, and this factor is the base raised to the lowest exponent. For example, x

^{2}is the common factor in the expression x

^{2}+ x

^{6}, and hence the expression can be rewritten as x

^{2}(1 + x

^{4}).

### Factoring a quadratic into binomials

**Description:**

This mini-lesson shows you how to factor a quadratic into binomials. As is the case in Algebra many times, the overview provided here in text might seem a little complicated, but don't worry -- it will be easy to follow once you hear the instructor explain it in the video provided. Some quadratics can be factored into two identical binomials. Such quadratics are called perfect square trinomials. As quadratic expression is the product of two binomials, factoring a quadratic means breaking the quadratic back into its binomial parts. Here factoring is done using the rule of LIOF (FOIL in reverse). A couple of general rules to keep in mind:

- The factoring of x
^{2}+ (a + b)x + ab will result into (x + a) (x + b). For example, the two factors of x^{2}+ 5x + 6 are (x + 2)(x + 3) - Another common type of algebraic factoring is called the difference of two squares: (x
^{2}– c^{2}) = (x + c) (x – c). For example: factors of x^{2}– 4 are (x + 2) (x – 2)

**Last Updated:**

**Subject(s):**- Mathematics
- Mathematics > Algebra

**Educational Level(s):**- Grades 9-10 / Ages 14-16
- Grades 11-12 / Ages 16-18

**Instructional Component Type(s):**- Curriculum: Lesson Plan

This mini-lesson shows you how to factor a quadratic into binomials. As is the case in Algebra many times, the overview provided here in text might seem a little complicated, but don't worry -- it will be easy to follow once you hear the instructor explain it in the video provided. Some quadratics can be factored into two identical binomials. Such quadratics are called perfect square trinomials. As quadratic expression is the product of two binomials, factoring a quadratic means breaking the quadratic back into its binomial parts. Here factoring is done using the rule of LIOF (FOIL in reverse). A couple of general rules to keep in mind:

- The factoring of x
^{2}+ (a + b)x + ab will result into (x + a) (x + b). For example, the two factors of x^{2}+ 5x + 6 are (x + 2)(x + 3) - Another common type of algebraic factoring is called the difference of two squares: (x
^{2}– c^{2}) = (x + c) (x – c). For example: factors of x^{2}– 4 are (x + 2) (x – 2)

### Factoring a quadratic using the perfect square method

**Description:**

In this mini-lesson you'll learn how to factor a quadratic using the perfect square method. In such cases, not only can the quadratic can be factored into two expressions, but the expressions are the same. If we try to explain it in text, here is the general rule -- if we have a quadratic equation in which first and last term are both perfect squares and middle term is two times the square root of the first and last terms multiplied, it simplifies the quadratic to a binomial product or just one binomial raised to the second power. Reading this explanation in text is confusing to many of us -- just click on the video of our instructor explaining it and you'll understand the concept much more easily. Note that perfect square trinomials are often expressions of one of the following forms:

- (x
^{2}+ 2ax + a^{2}), which is the same as (x + a)^{2} - (x
^{2}– 2ax + a^{2}), which is the same as (x – a)^{2}

**Last Updated:**

**Subject(s):**- Mathematics
- Mathematics > Algebra

**Educational Level(s):**- Grades 9-10 / Ages 14-16
- Grades 11-12 / Ages 16-18

**Instructional Component Type(s):**- Curriculum: Lesson Plan

In this mini-lesson you'll learn how to factor a quadratic using the perfect square method. In such cases, not only can the quadratic can be factored into two expressions, but the expressions are the same. If we try to explain it in text, here is the general rule -- if we have a quadratic equation in which first and last term are both perfect squares and middle term is two times the square root of the first and last terms multiplied, it simplifies the quadratic to a binomial product or just one binomial raised to the second power. Reading this explanation in text is confusing to many of us -- just click on the video of our instructor explaining it and you'll understand the concept much more easily. Note that perfect square trinomials are often expressions of one of the following forms:

- (x
^{2}+ 2ax + a^{2}), which is the same as (x + a)^{2} - (x
^{2}– 2ax + a^{2}), which is the same as (x – a)^{2}

###
Factoring 3^{rd} degree polynomial

**Description:**

In this mini-lesson you'll learn with the help of several examples how to factor 3

^{rd}degree polynomial into 2

^{nd}degree polynomial and 1

^{st}degree polynomial factor. As you know, if you write a polynomial as the product of two or more polynomials, you have factored it. It is fairly common to come across certain interesting forms of third degree polynomials, and here are a few rules to keep in mind in factoring them:

RULE 1: a

^{3}+ b

^{3}= (a + b) (a

^{2}– ab + b

^{2})

RULE 2: a

^{3}– b

^{3}= (a – b)(a

^{2}+ ab + b

^{2})

RULE 3: (x + y + z)(x

^{2}+ y

^{2}+ z

^{2}– xy – yz – zx) = x

^{3}+ y

^{3}+ z

^{3}– 3xyz.

**Last Updated:**

**Subject(s):**- Mathematics
- Mathematics > Algebra

**Educational Level(s):**- Grades 9-10 / Ages 14-16
- Grades 11-12 / Ages 16-18

- high
- 9th
- 10th
- secondary
- freshman
- sophomore
- teen
- 11th
- 12th
- senior

**Instructional Component Type(s):**- Curriculum: Lesson Plan

In this mini-lesson you'll learn with the help of several examples how to factor 3

^{rd}degree polynomial into 2

^{nd}degree polynomial and 1

^{st}degree polynomial factor. As you know, if you write a polynomial as the product of two or more polynomials, you have factored it. It is fairly common to come across certain interesting forms of third degree polynomials, and here are a few rules to keep in mind in factoring them:

RULE 1: a

^{3}+ b

^{3}= (a + b) (a

^{2}– ab + b

^{2})

RULE 2: a

^{3}– b

^{3}= (a – b)(a

^{2}+ ab + b

^{2})

RULE 3: (x + y + z)(x

^{2}+ y

^{2}+ z

^{2}– xy – yz – zx) = x

^{3}+ y

^{3}+ z

^{3}– 3xyz.