Factoring Completely

Factoring is an important and useful concept in algebraic mathematics. It is used to simplify the given expression into simpler terms. When we simplify the given expression into its factors then this process is called factoring. In this article, we will learn in deep about the concept of "Factoring Completely".

What is Factoring?

Factoring is the process of breaking down a given algebraic expression into simpler parts known as factors. When these factors are multiplied it gives back the original expression.

For example- If we are given an algebraic expression:

x² - 4x + 4

If we factor the above expression then it becomes (x - 2) (x - 2) where (x -2) are its factor

When we multiply (x -2) × (x - 2) = x² - 4x + 4

Basic Factoring Methods

Some of the common factoring methods are:

• GCF
• Factoring by Grouping
• Factoring Trinomials (AC Method)
• Factoring Difference of Squares
• Factoring Perfect Square Trinomials
• Factoring Sum and Difference of Cubes
• Factoring by Substitution

Factoring Out the Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest term that divides all terms in an expression. Factoring out GCF from expression is simple and first step to factor the given expression.

Example: Factor out the GCF from the expression 12x²y - 8xy² + 16xy.

Solution:

In the given expression: 12x²y - 8xy² + 16xy.

Identify the GCF of the terms: 4xy

Factor out the GCF: 4xy(3x - 2y + 4)

Factoring by Grouping

In this method we factor the given expression by rearranging terms in an expression and factoring common factors from groups of terms. This method is very useful when we are dealing with polynomials which has total four-terms.

Example: Factor the given expression by grouping: x² + 3x² - 4x - 12.

Solution:

Group the terms: (x³ + 3x²) − (4x + 12)

Factor out the GCF from each group: x²(x + 3) - 4(x + 3)

Factor out the common binomial factor: (x +3) (x² - 4)

If we factor it further: (x +3) (x +2) ( x -2)

Factoring Trinomials (AC Method)

A trinomial is an expression which has three terms. Trinomial is most common type of expression in algebra mathematics. The most common trinomial in mathematic is quadratic equation which can be factored by using several methods.

Example: Factor the given trinomial: x² + 5x + 6.

Solution:

x² + 5x + 6 = 0

Now, write the middle term (5x) as a sum of two terms whose coefficients add up to 5:

x² + 2x + 3x + 6 = 0

Group the terms of the equation:

(x² + 2x) + (3x + 6) = 0

Factor out the common factors from each group:

x(x + 2) + 3(x + 2) = 0

Factor out the common binomial factor:

(x + 2) (x + 3) = 0

Factoring Difference of Squares

The difference of square is a type of binomial problem. It involves a specific type of binomial that can be factored using the formula.

The formula is: a² - b² = (a + b) (a − b)

Let's consider an example for better understanding:

Example: Factor x² - 25.

Solution:

Given binomial expression: x² - 25

Now, we identify it as the difference of square:

(x)² - (5)²

(x - 5) (x + 5)

Factoring Perfect Square Trinomials

A perfect square trinomial is a three term given expression. A perfect square trinomial is facotrd by using formulas. Two most common used formulas are:

(a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b²

Example 1: Factor: x² + 6x + 9

Solution:

Identify, the given expression: x² + 6x + 9, it is a perfect trinomial

Now, we factorn using the formula: (a + b)² = a² + 2ab + b²

(x)²+ 2 × 3 × x + (3)²

Now factors of this trinomial: (x + 3)²

Example 2: Factor: x² - 10x + 25

Solution:

Identify, the given expression: x² - 10x + 25, it is a perfect trinomial

Now, we factorn using the formula: (a - b)² = a² - 2ab + b²

(x)² - 2 × 5 × x + (5)²

Now factors of this trinomial: (x - 5)²

Factoring Sum and Difference of Cubes

The two main formula for addition and difference of cubes can be used in to factor expression.

(a)³ + (b)³ = (a + b) (a² - ab + b²)

(a)³ - (b)³ = (a - b) (a² + ab + b²)

Example: Factor x³ - 27.

Solution:

Try to identify pattern in this question

x³ - 27 = (x)³ - (3)²

The formula is: (a)³ - (b)³ = (a - b) (a² + ab + b²)

Apply the formula: (x − 3) (x² + 3x + 9)

Factoring by Substitution

In factoring higher degree polynomials, we can use substitution method. this can be easily understood by taking an example.

Example: Factor: x⁴ - 5x² + 6

Solution:

Let y = x² substitute this in given expression: x⁴ - 5x² + 6

Now the above expression becomes y² -5y + 6

Factor y² - 5y + 6 into (y -2) (y - 3)

now substitute back y = x²

(x² - 2) (x² - 3)

Practice Questions on Factoring

Q1: Factor the expression by grouping: x³ + 2x² - 3x - 6.

Solution:

Group the terms: (x³ + 2x² ) - (3x + 6)

Factor out the GCF from each group: x² (x + 2) - 3(x + 2)

Factor out the common binomial factor: (x +2) (x² - 3

Q2: Factor the given trinomial: x² + 7x + 12.

Solution:

given expression: x²+ 7x + 12

now, write the middle term as a sum of two terms whose coefficients add up to 7:

x² + 3x + 4x + 12

Group the terms: (x² + 3x) + (4x + 12) = 0

Factor out the common factors from each group: x (x + 3) + 4 (x + 3) = 0

Factor out the common binomial factor: (x + 3) (x + 4) = 0

Q3: Factor 4x² - 49.

Solution:

Identify the given expression: 4x² - 49

as difference of square: (2x)² - (7)²

Apply the formula: a² - b² = (a + b) (a - b)

(2x + 7) ( 2x - 7)

Q4: Factor: x⁶ - 7x³ + 12

Solution:

Given expression: x⁶ - 7x³ + 12

Let y = x³ and sunstiute in above expression

y² - 7y + 12

now, facotr the quadratic:

( y - 3) ( y - 4)

Q5: Factor x³ + 8.

Solution:

Given expression: x³ + 8

Identify the sum of cubes: (x)³ + (2)³

We, apply the formula a³ + b³ = (a + b) (a² - ab + b²)

Applying the formula: (x)³ + (2)³ = (x + 4) (x² - 2x + 4)

Q6: Factor the given expression by grouping: 3x³ + 6x² -2x -4

Solution:

First, we group the terms: (3x³ + 6x²) - (2x + 4)

Factor out the GCF from each group: 3x² (x + 2) - 2(x + 2)

Factor out the common binomial factor: (x + 2) (3x² - 2)

Q7: Factor the given expression: 16x² + 24x + 9.

Solution:

Identify the given expression as the perfect square trinomial:

(4x)² + 2 × 4x × 3 + (3)²

Factor using the formula

(a+b² = a² + 2ab + b²

(4x + 3)²

Q8: Factor the expression: x³ + 4x² -x -4

Solution:

Group the terms: (x³ + 4x² ) - (x + 4)

Factor out the GCF from each group: x² (x + 4) - 1(x + 4)

Factor out the common binomial factor: (x + 4) (x² - 1)

If we further factor the expression: (x + 4) (x − 1) (x + 1)

Conclusion

Factoring is the process of simplifying the complex expression into simple parts known as its factors. After breaking down polynomials and other algebraic expressions into their basic factors,we make it easier to solve equations, simplify expressions, and understand the structure of mathematical problems. Factoring is very important concept of mathematics as it laid down foundation for advanced topics in algebra and calculus.

Read More,

• Polynomial
• Factoring Polynomial
• Quadratic Equation
• Standard Form of Quadratic Equation

FAQs on Factoring Completely

What is a perfect square trinomial?

A perfect square trinomial is an expression that can be written as (a - b)² or (a + b)²

• (a + b)² = a² + 2ab + b²
• (a - b)² = a²- 2ab + b²

How we can factor a polynomial with four terms?

We can factor a polynomial with four terms by using grouping method. We group the terms into pairs, factor out the GCF from each pair, and then factor out the common binomial factor from the resulting groups.

What are some real-world applications of factoring polynomials?

Factoring polynomials has real-world applications in various fields such as engineering, physics, and computer science.

Name some techniques that are used for factoring polynomials of high degree?

For factoring high-degree polynomials, advanced techniques such as synthetic division, numerical methods, and computer algebra systems (CAS) are used.

B

bug8wdqo

Follow

Improve

Article Tags :

• Mathematics
• School Learning
• Maths