Factoring Polynomials
Factoring Polynomials means decomposing the given polynomial into a product of two or more polynomials using prime factorization. Factoring polynomials help in simplifying the polynomials easily. The first step is to write each term of the larger expression as a product of its factors. As a second step, the common factors across the terms are taken out in common to create the required factors. Let's discuss the methods of factoring polynomials, and some of the important concepts related to factoring polynomials: remainder theorem, factor theorem, GCF, long division.
What is Factoring of Polynomials?
The process of factoring polynomials involves expressing the polynomial as the product of its factors. Factoring polynomials help in finding the values of the variables of the given expression or to find the zeros of the polynomial expression. A polynomial is of the form axn + bxn - 1 + cxn - 2+ .........px + q, which can be factorized using numerous methods: grouping, using identities and substituting.
Here in this polynomial, the exponent of x is n and it has n factors. The number of factors is equal to the degree of the variable in the polynomial expression. Higher degree polynomials are reduced to a simpler lower degree, linear or quadratic expressions to obtain the required factors. Factoring polynomials can be understood with the help of a simple example. The quadratic polynomial x2 + x(a + b) + ab can be factorized as (x + a)(x + b).
Example
Here's an example for solving polynomial factorization:
"Factor the polynomial \( x^2 - 5x + 6 \)."
The formula to factor a quadratic polynomial is:
\( ax^2 + bx + c = (px + q)(rx + s) \)
For the polynomial \( x^2 - 5x + 6 \), we have the following values:
\( a = 1 \), \( b = -5 \), and \( c = 6 \).
We are looking for two numbers that multiply to \( c = 6 \) and add up to \( b = -5 \).
The numbers that satisfy this are \( -2 \) and \( -3 \), because:
\( (-2) \times (-3) = 6 \) and \( (-2) + (-3) = -5 \).
Therefore, we can factor the polynomial as:
\( x^2 - 5x + 6 = (x - 2)(x - 3) \)
The factorized form of the polynomial is: \( (x - 2)(x - 3) \)
This demonstrates how polynomial factorization can be used to break down a quadratic polynomial into its factors.
| Polynomial | Factorization Formula | Factors | Coefficients |
|---|---|---|---|
| \( x^2 - 5x + 6 \) | \( x^2 - 5x + 6 = (x - 2)(x - 3) \) | Factors: \( (x - 2)(x - 3) \) | Coefficients: \( a = 1 \), \( b = -5 \), \( c = 6 \) |
| \( x^2 + 7x + 10 \) | \( x^2 + 7x + 10 = (x + 5)(x + 2) \) | Factors: \( (x + 5)(x + 2) \) | Coefficients: \( a = 1 \), \( b = 7 \), \( c = 10 \) |
| \( x^2 - 3x - 18 \) | \( x^2 - 3x - 18 = (x - 6)(x + 3) \) | Factors: \( (x - 6)(x + 3) \) | Coefficients: \( a = 1 \), \( b = -3 \), \( c = -18 \) |
| \( x^2 + 2x - 35 \) | \( x^2 + 2x - 35 = (x + 7)(x - 5) \) | Factors: \( (x + 7)(x - 5) \) | Coefficients: \( a = 1 \), \( b = 2 \), \( c = -35 \) |
| \( x^2 - 4x - 21 \) | \( x^2 - 4x - 21 = (x - 7)(x + 3) \) | Factors: \( (x - 7)(x + 3) \) | Coefficients: \( a = 1 \), \( b = -4 \), \( c = -21 \) |
