Limits and Continuity: Background You’ll Need 1

  • Factor and simplify polynomials and rational expressions

Factor Polynomials

Factoring is central to simplifying expressions, solving equations, and understanding polynomial behavior. Factoring involves breaking down expressions into simpler, constituent parts. A key step in this process is identifying the greatest common factor (GCF), which simplifies polynomials by dividing out commonalities and reducing complexity.

Greatest Common Factor

The greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. 

[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].

The GCF of polynomials works the same way.

[latex]4x[/latex] is the GCF of [latex]16x[/latex] and [latex]20{x}^{2}[/latex] because it is the largest polynomial that divides evenly into both [latex]16x[/latex] and [latex]20{x}^{2}[/latex].

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. 

greatest common factor (GCF) of a polynomial

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

To make it less challenging to find this GCF of the polynomial terms, first look for the GCF of the coefficients, and then look for the GCF of the variables.

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.

The distributive property allows us to multiply a number by a sum or difference inside parentheses and add or subtract the results. Conversely, when we see a common factor shared by all terms, we can factor it out, effectively reversing the distributive process.

  • Using the distributive property: [latex]a\left(b+c\right)=ab+ac[/latex].
  • Factoring out a common factor: [latex]ab+ac=a\left(b+c\right)[/latex].

This principle shows us that multiplication distributed across a sum can be “undone” through factoring, revealing the GCF and the remaining terms of the polynomial.

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

Watch this video to see more examples of how to factor the GCF from a trinomial.

You can view the transcript for “Ex 2: Identify GCF and Factor a Trinomial” here (opens in new window).

Factoring Quadratic Trinomials with a Leading Coefficient of [latex]1[/latex]

When factoring polynomials, starting with the greatest common factor (GCF) is standard. However, the GCF is not always the key to simplification, particularly for polynomials without a common factor.

For instance, the quadratic trinomial [latex]{x}^{2}+5x+6[/latex] has a GCF of 1, but it can be written as the product of the factors [latex]\left(x+2\right)[/latex] and [latex]\left(x+3\right)[/latex].

To factor trinomials like [latex]{x}^{2}+bx+c[/latex], find two numbers that multiply to [latex]c[/latex] and add up to [latex]b[/latex].

The trinomial [latex]{x}^{2}+10x+16[/latex] can be factored using the numbers [latex]2[/latex] and [latex]8[/latex], because [latex]2 \times 8 =16[/latex] and [latex]2 + 8 = 10[/latex]. The trinomial can be rewritten as the product of [latex]\left(x+2\right)[/latex] and [latex]\left(x+8\right)[/latex].

factoring quadratic trinomials with a leading coefficient of [latex]1[/latex]

A trinomial of the form [latex]{x}^{2}+bx+c[/latex] can be written in factored form [latex]\left(x+p\right)\left(x+q\right)[/latex] where [latex]p \times q=c[/latex] and [latex]p+q=b[/latex].

It’s a common misconception that all trinomials can be broken down into binomial factors, but this isn’t always the case. While many polynomials can be factored in this way, revealing a product of simpler binomials, there are instances where a trinomial is prime and cannot be factored further using real numbers

How To: Factoring a Trinomial of the Form [latex]{x}^{2}+bx+c[/latex]

  1. Identify all factor pairs of [latex]c[/latex].
  2. Find the factor pair where the sum equals [latex]b[/latex].
  3. Write the trinomial as the product of two binomials, [latex]\left(x+p\right)\left(x+q\right)[/latex].

To verify the accuracy of our factorization, we can employ the FOIL method, which stands for First, Outer, Inner, Last. This technique allows us to multiply two binomials and ensures that our factorization is correct. If the expanded expression matches the original polynomial, our factorization is verified.

The FOIL method is a process used to multiply two binomials. The acronym FOIL stands for:

  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outermost terms in the product.
  • Inner: Multiply the innermost terms.
  • Last: Multiply the last terms in each binomial.

After applying the FOIL method, combine like terms to get the final expanded expression.

Factor [latex]{x}^{2}+2x - 15[/latex].


Simplify Rational Expressions

A rational expression is formed by dividing one polynomial by another. To simplify these expressions, we use fraction properties, particularly focusing on reducing common factors between the numerator and denominator.

Here’s the process:

[latex]\text{Original Expression: }\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[/latex]

Step 1: Factor both the numerator and denominator.

[latex]\text{Factorized Form: }\frac{{(x+4)}{(x+4)}}{\left(x+4\right)\left(x+7\right)}[/latex]
Step 2: Cancel out common factors.

[latex]\text{Simplified Expression: }\frac{x+4}{x+7}[/latex].

By removing the common factor of [latex]x+4[/latex], we’ve simplified the rational expression to its reduced form.

How To: Simplify a Rational Expression

  1. Identify the Polynomials: Recognize the numerator and denominator as separate polynomials.
  2. Factor Completely: Break down both the numerator and the denominator into their prime factors.
  3. Cancel Common Factors: Look for and cancel out any factors that appear in both the numerator and the denominator.
  4. Write the Simplified Expression: After canceling the common factors, write down what remains.

Simplify [latex]\frac{{x}^{2}-9}{{x}^{2}+4x+3}[/latex].

Can the [latex]{x}^{2}[/latex] term be cancelled in the above example?

No. A factor is an expression that is multiplied by another expression. The [latex]{x}^{2}[/latex] term is not a factor of the numerator or the denominator.

Simplify [latex]\frac{x - 6}{{x}^{2}-36}[/latex].

 


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