Factoring Polynomials: Learn It 1

  • Factor polynomial expressions using the Greatest Common Factor (GCF) and by grouping to simplify expressions.
  • Factor trinomials and perfect square trinomials into binomials.
  • Break down expressions like differences of squares and cubic equations into their simpler factors.
  • Use specific methods to factor expressions that contain fractional or negative exponents.

Factoring polynomials is like breaking down a complex mathematical phrase into simpler parts that are easier to handle. Many polynomial expressions, which are combinations of numbers and variables raised to various powers, can often be rewritten in simpler forms by factoring. This process involves finding numbers or expressions that, when multiplied together, produce the original polynomial.

Factoring is not just a mathematical trick; it’s a fundamental tool that helps in solving equations, simplifying expressions, and understanding algebraic structures more deeply.

Factoring the Greatest Common Factor (GCF)

When factoring a polynomial expression, our first step is to check to see if each term contains a common factor. If so, we factor out the greatest amount we can from each term. Finding and factoring out a greatest common factor from a polynomial is the first skill involved in factoring polynomials.

Recall that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For example, [latex]4[/latex] is the GCF of [latex]16[/latex] and [latex]20[/latex] because it is the largest number that divides evenly into both [latex]16[/latex] and [latex]20[/latex].

GCF of Polynomials

The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.

 

Identifying and factoring out the GCF involves recognizing the highest degree of any common variables and the largest factor common to all numerical coefficients.

To factor out a GCF from a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property “backwards” to rewrite the polynomial in a factored form.

Distributive Property: [latex]a\left(b+c\right)=ab+ac[/latex].

Backward: Let’s factor [latex]a[/latex] out of [latex]ab+ac[/latex].

[latex]ab+ac=a\left(b+c\right)[/latex].

We have seen that we can distribute a factor over a sum or difference. Now we see that we can “undo” the distributive property with factoring.

How To: Given a polynomial expression, factor out the greatest common factor

  1. Identify the GCF of the coefficients.
  2. Identify the GCF of the variables.
  3. Combine to find the GCF of the expression.
  4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.
  5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.
Factor [latex]25b^{3}+10b^{2}[/latex].

Solution

  1. Identify the GCF of the Coefficients: The coefficients are [latex]25[/latex] and [latex]10[/latex]. The GCF of [latex]25[/latex] and [latex]10[/latex] is [latex]5[/latex].
  2. Identify the GCF of the Variables: The terms include [latex]b^3[/latex] and [latex]b^2[/latex]. The GCF of [latex]b^3[/latex] and [latex]b^2[/latex] is [latex]b^2[/latex](the term with the lowest power).
  3. Combine the GCFs: Combine the GCF of the coefficients with the GCF of the variables. The overall GCF is [latex]5b^2[/latex].
  4. Determine what the GCF needs to be multiplied by to obtain each term in the expression:
    • [latex]25b^3 = 5b^2 \cdot 5b[/latex]
    • [latex]10b^2 = 5b^2 \cdot 2[/latex]
  5. Write the Factored Form:

[latex]25b^{3}+10b^{2} = 5b^2 \cdot 5b + 5b^2 \cdot 2 = 5b^2(5b+2)[/latex]

Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[/latex].

The GCF may not always be a monomial. Here is an example of a GCF that is a binomial.

Factor [latex]x\left({b}^{2}-a\right)+6\left({b}^{2}-a\right)[/latex] by pulling out the GCF.