Factoring Higher Degree Polynomials

Factoring Higher-Degree Polynomials (those of at least three degrees) is an important concept in algebra-based math. this includes separating a complex polynomial into less complex, multipliable parts called factors. These factors are more straightforward to work with, particularly for solving equations or simplifying expressions.

What Are Polynomials?

Polynomials are logarithmic expressions that contain factors raised to non-negative whole number powers and their coefficients. They are key in polynomial math and analytics and they are used to display different real-life scenarios.

A polynomial is generally expressed in the following form:

P(x) = a_n xⁿ + a_n-₁x^n-1 + . . . . + a₁x + a₀

Where,

• x represents the variable.
• a_n, a_n-1, . . ., a₀ are constants called coefficients.
• n is a non-negative integer called the degree of the polynomial.

Read More about Factoring Polynomial.

What Are Higher Degree Polynomials?

Higher degree polynomials are polynomials that have a degree of higher value such as three (cubic), fiver (quintic), or even higher degree than that.

The higher degree polynomials are important and used widely in the field of mathematics and engineering because they can be used for modeling the complex relationships between data and it provide solutions to real-world data etc.

Example of higher degree polynomials:

Higher degree polynomials are simple to represent in mathematics, represented as follows:

• A cubic polynomial can be represented as x³ + 2x² - 5x + 7
• Similarly, a quartic polynomial can be represented as follows: 2x⁴ - 3x³ + x - 6.

This is how we can represent a higher degree polynomial.

How to Factorize Higher Degree Polynomials?

Factorizing the higher degree polynomials are a challenging task but there are some common methods which are used in mathematics for the factorizing of higher degree polynomials such as following:

Factoring by Grouping

We can factorize the higher degree polynomials by grouping them, for this we can solve a given higher degree polynomial as following:

x³ + 3x² + 2x + 6 = (x³ + 3x²) + (2x + 6) = x²(x+3) +2(x+3) = (x²+ 2)(x+3)

Quadratic Formula

If we can manage to reduce the given polynomial to a quadratic form then we can easily factor it using the quadratic formula.

Special Formulas

We can also use some other known formulas such as sum or difference of cubes or squares etc. one such formula is following:

a² − b² = (a − b)(a + b)

a³ + b³ = (a + b)(a²- ab + b²)

a³ − b³= (a − b)(a² + ab + b²)

a⁴ − b⁴ = (a² + b²)(a − b)(a + b)

a⁶ + b⁶ = (a² + ab + b²)(a⁴ − a²b² + b⁴)

Examples Related to Factorize Higher Degree Polynomials

Now that we have understood what are higher degree polynomial and different methods of factorizing higher degree polynomials, lets take a look at example to understand how higher degree polynomials are factorized:

Example 1: Factorize x⁴ - 16.

Solution:

Step 1: From the question, we can recognize that this as difference of squares, and it can be then written as:

x⁴ - 16 = (x²)² - 4²

Step 2: We can now factor the equation using the square difference formula:

x⁴ - 16 = (x² - 4)(x² + 4)

Step 3: We can factorize it further as following:

x² - 4 = (x - 2)(x + 2)

Step 4: Final factorized higher degree polynomial will be:

x⁴ - 16 = (x-2)(x+2)(x²+4)

Example 2 : Solve the sum of cubes in higher degree polynomial:Factorize x³ + 27

Solution:

Following are the steps required for solving this higher degree polynomial:

Step 1: Find pattern, here it can be seen that it is a sum of cubes so it can be written as:

x³ + 27 = x³ + 3³

Step 2: Use the sum of cubes formula (given below) to solve the above equation.

a³ + b³ = (a + b) (a² - ab + b²)

The above equation can be represented as following by using the sum of cubes formula:

x³ + 27 = (x + 3) (x² - 3x + 9)

This will be the final factorized form of the given higher degree polynomial.

Practice Questions on Factoring Higher Degree Polynomials

Following are some practice questions on the factoring of higher degree polynomials to understand it better:

Question 1 : Factor the polynomial: x³ + 6x² + 11x + 6

Question 2 - Factor the polynomial: 2x⁴ - 18x² + 8

Question 3 - Factor the polynomial: x⁵ - x³

Question 4 - Factor the polynomial: x⁴ + 4x² + 16

Question 5 - Factor the polynomial: x³ - 8

Conclusion

Factoring of higher degree polynomials are important concept in the field of mathematics because it helps us in various mathematical fields such as modeling and data analysis etc. some common methods which have been discussed in this article include grouping, quadratic formula and other special formulas for solving the higher degree polynomials.

Related article

• Factoring Polynomials
• Factorization of polynomial
• Factorization of polynomials practice problem
• Factors Theorem