How to Factor a Polynomial
Understanding Polynomials and their Components
Polynomials are expressions made up of one or more terms that involve variables and coefficients. The degree of a polynomial is determined by the highest exponent of the variable. For example, in the polynomial 4x^3 + 2x^2 – 6x + 1, the degree is 3.
There are several key components of a polynomial that are important to understand when factoring. The terms of a polynomial are separated by addition or subtraction signs. Each term consists of a coefficient and a variable raised to a certain power. The constant term, if present, is simply a number without a variable.
In order to factor a polynomial, it is necessary to identify its factors. The factors of a polynomial are the expressions that can be multiplied together to obtain the original polynomial. By factoring a polynomial, it is possible to simplify it and solve equations that involve the polynomial.
Factoring by Greatest Common Factor (GCF)
One of the simplest and most common methods of factoring polynomials is to use the greatest common factor (GCF). The GCF is the largest factor that divides all the terms of a polynomial evenly.
To factor a polynomial using the GCF, start by identifying the GCF of all the terms. Then, divide each term by the GCF and write the result as a product of the GCF and the quotient.
For example, consider the polynomial 12x^2 + 24x. The GCF of the terms is 12x. Divide each term by 12x to obtain:
12x^2 / 12x = x
24x / 12x = 2
Thus, the polynomial can be factored as:
12x^2 + 24x = 12x(x + 2)
Note that the GCF can also include variables. In this case, the GCF would be the variable with the lowest exponent that appears in all the terms.
Factoring Trinomials using Different Techniques
A trinomial is a polynomial with three terms. Factoring trinomials can be done using different techniques depending on the structure of the trinomial.
Factoring a trinomial of the form x^2 + bx + c:
To factor a trinomial of the form x^2 + bx + c, find two numbers that multiply to c and add up to b. Then, write the trinomial as a sum of these two terms and factor by grouping.
For example, consider the trinomial x^2 + 7x + 10. Find two numbers that multiply to 10 and add up to 7. The numbers are 2 and 5. Thus, the trinomial can be written as:
x^2 + 7x + 10 = x^2 + 2x + 5x + 10
Factor by grouping:
x(x + 2) + 5(x + 2) = (x + 2)(x + 5)
Factoring a trinomial of the form ax^2 + bx + c:
To factor a trinomial of the form ax^2 + bx + c, find two numbers that multiply to a*c and add up to b. Then, write the trinomial as a sum of these two terms and factor by grouping.
For example, consider the trinomial 6x^2 + 5x – 6. Find two numbers that multiply to 6*(-6) = -36 and add up to 5. The numbers are 8 and -3. Thus, the trinomial can be written as:
6x^2 + 5x – 6 = 6x^2 + 8x – 3x – 6
Factor by grouping:
2x(3x + 4) – 3(3x + 4) = (2x – 3)(3x + 4)
Factoring a trinomial of the form ax^2 + bx + c by completing the square:
To factor a trinomial of the form ax^2 + bx + c by completing the square, add and subtract (b/2a)^2 inside the trinomial. Then, write the trinomial as a square of a binomial and simplify.
For example, consider the trinomial x^2 + 4x + 3. Add and subtract (4/2)^2 = 4 inside the trinomial to obtain:
x^2 + 4x + 4 – 4 + 3 = (x + 2)^2 – 1
Thus, the trinomial can be factored as:
x^2 + 4x + 3 = (x + 2)^2 – 1 = (x + 1)(x + 3)
Factoring Quadratic Expressions using the Quadratic Formula
The quadratic formula is a formula that can be used to solve any quadratic equation of the form ax^2 + bx + c = 0. It can also be used to factor quadratic expressions by finding the roots of the equation and writing it as a product of two linear factors.
The quadratic formula is:
x = (-b ± sqrt(b^2 – 4ac)) / 2a
To factor a quadratic expression using the quadratic formula, first find the values of a, b, and c. Then, use the quadratic formula to find the roots of the equation. Finally, write the quadratic expression as a product of two linear factors with the roots as the solutions.
For example, consider the quadratic expression 3x^2 + 7x + 2. Find the values of a, b, and c:
a = 3, b = 7, c = 2
Use the quadratic formula to find the roots:
x = (-7 ± sqrt(7^2 – 432)) / 2*3
x = (-7 ± sqrt(49 – 24)) / 6
x = (-7 ± sqrt(25)) / 6
x = (-7 ± 5) / 6
Thus, the roots are x = -2/3 and x = -1.
Write the quadratic expression as a product of two linear factors:
3x^2 + 7x + 2 = 3(x + 2/3)(x + 1)
Factoring Higher Degree Polynomials using Synthetic Division and Long Division
When factoring polynomials with a degree higher than 2, it can be difficult to identify the factors using the methods discussed above. Synthetic division and long division are two methods that can be used to factor higher degree polynomials.
Synthetic Division:
Synthetic division is a shorthand method of polynomial division that can be used to factor polynomials of degree 2 or higher. The process involves dividing the polynomial by a linear factor and using the remainder to find the next factor.
For example, consider the polynomial x^3 – 2x^2 – x + 2. Divide by the linear factor x – 1 using synthetic division:
1 | 1 -2 -1 2
| 1 -1 0
-------------
1 -3 -2 2
Thus, the polynomial can be factored as:
x^3 – 2x^2 – x + 2 = (x – 1)(x^2 – 3x – 2)
Long Division:
Long division is a method of polynomial division that can be used to factor polynomials of any degree. The process involves dividing the polynomial by a linear factor and using the quotient and remainder to find the next factor.
For example, consider the polynomial x^4 – 3x^3 – 4x^2 + 12x + 4. Divide by the linear factor x – 2 using long division:
x^3 + x^2 - 2x - 2
------------------------
x - 2 | x^4 - 3x^3 - 4x^2 + 12x + 4
x^4 - 2x^3
------------
-x^3 - 4x^2
-x^3 + 2x^2
------------
-6x^2 + 12x
-6x^2 + 12x
------------
0
Thus, the polynomial can be factored as:
x^4 – 3x^3 – 4x^2 + 12x + 4 = (x – 2)(x^3 + x^2 – 2x – 2)
