The Limit Laws section will introduce yet another way to calculate a limit, using limit laws. Basically, calculating limits of functions algebraically will be the topic of focus in the section. Several techniques that are used to simplify functions are sometimes needed to successfully calculate a function’s limit. Here we will review factoring, simplifying rational expressions, rationalizing radical expressions, and simplifying complex rational expressions.

Factor 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]. 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].

Finding and factoring out a GCF from a polynomial is the first skill involved in factoring polynomials.

Factor a GCF out of a Polynomial

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. 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. Recall that the distributive property of multiplication over addition states that a product of a number and a sum is the same as the sum of the products.

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).

Factor a Trinomial with Leading Coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial [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].

Trinomials of the form [latex]{x}^{2}+bx+c[/latex] can be factored by finding two numbers with a product of [latex]c[/latex] and a sum of [latex]b[/latex]. The trinomial [latex]{x}^{2}+10x+16[/latex], for example, can be factored using the numbers [latex]2[/latex] and [latex]8[/latex] because the product of these numbers is [latex]16[/latex] and their sum is [latex]10[/latex]. The trinomial can be rewritten as the product of [latex]\left(x+2\right)[/latex] and [latex]\left(x+8\right)[/latex].

Factor by Grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial [latex]2{x}^{2}+5x+3[/latex] can be rewritten as [latex]\left(2x+3\right)\left(x+1\right)[/latex] using this process. We begin by rewriting the original expression as [latex]2{x}^{2}+2x+3x+3[/latex] and then factor each portion of the expression to obtain [latex]2x\left(x+1\right)+3\left(x+1\right)[/latex]. We then pull out the GCF of [latex]\left(x+1\right)[/latex] to find the factored expression.

In the next video, we see another example of how to factor a trinomial by grouping.

You can view the transcript for “Factor a Trinomial in the Form ax^2+bx+c Using the Grouping Technique” here (opens in new window).

Factor a Difference of Squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

[latex]\\[/latex]

We can use this equation to factor any differences of squares.

Watch this video to see another example of how to factor a difference of squares.

You can view the transcript for “Ex: Factor a Difference of Squares” here (opens in new window).

Factor the Sum and Difference of Cubes

Now we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

[latex]\\[/latex]

Similarly, the sum of cubes can be factored into a binomial and a trinomial but with different signs.

[latex]\\[/latex]

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: Same Opposite Always Positive. For example, consider the following example.

The sign of the first 2 is the same as the sign between [latex]{x}^{3}-{2}^{3}[/latex]. The sign of the [latex]2x[/latex] term is opposite the sign between [latex]{x}^{3}-{2}^{3}[/latex]. And the sign of the last term, 4, is always positive.

In the following two video examples, we show more binomials that can be factored as a sum or difference of cubes.

You can view the transcript for “Ex 1: Factor a Sum or Difference of Cubes” here (opens in new window).

You can view the transcript for “Ex 3: Factor a Sum or Difference of Cubes” here (opens in new window).

Simplify Rational Expressions

The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.

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

[latex]\\[/latex]We can factor the numerator and denominator to rewrite the expression as [latex]\frac{{\left(x+4\right)}^{2}}{\left(x+4\right)\left(x+7\right)}[/latex]

Rationalize Radical Expressions

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator.

We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by a form of 1 that will eliminate the radical.

For a denominator containing a binomial where at least one of the terms is a square root, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign in the middle of the binomial. If the denominator is [latex]a+b\sqrt{c}[/latex], then the conjugate is [latex]a-b\sqrt{c}[/latex].

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\dfrac{a}{\dfrac{1}{b}+c}[/latex] can be simplified by rewriting the numerator as the fraction [latex]\dfrac{a}{1}[/latex] and combining the expressions in the denominator as [latex]\dfrac{1+bc}{b}[/latex]. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\dfrac{a}{1}\cdot \dfrac{b}{1+bc}[/latex] which is equal to [latex]\dfrac{ab}{1+bc}[/latex].