Lesson Video: Factoring Trinomials Mathematics

Video Transcript

In this video, we will learn how to factor trinomials into a product of two binomials.

We begin by recalling the definitions of monomials, binomials, and trinomials. A monomial is a product of numbers and powers of variables. A binomial expression is the sum or difference of two monomials. A trinomial expression is the sum or difference of three monomials. An example of a monomial expression is negative five 𝑥 squared 𝑦. An example of a binomial expression is three 𝑥 squared plus seven. An example of a trinomial expression is two 𝑎 squared minus two 𝑎𝑏 plus three 𝑏.

When we list the factors of a number, we can write the number as a product of its factors. For example, we may write 20 equal to two multiplied by 10. The same principle is true when we factor algebraic expressions. The focus of this video is writing trinomials as the product of two binomial factors.

In general, when we multiply two binomials, we initially obtain four terms, created by multiplying each term in one binomial by each term in the other. If the two binomials have the same algebraic structure, then we are able to combine one pair of like terms, leading to a trinomial. Each of the examples we consider in this video are of this type. We first demonstrate the process of factoring an expression with a common binomial term that is already in a partially factored form.

Fully factor the expression 𝑥 multiplied by 𝑥 plus three plus two multiplied by 𝑥 plus three.

Upon inspection, we observe that the two parts of this expression share a common binomial factor of 𝑥 plus three. We can, therefore, factor by this shared binomial. In the first part of the expression, this binomial is multiplied by 𝑥. And in the second part, it is multiplied by two. Hence, overall, it is multiplied by 𝑥 plus two. And so, 𝑥 multiplied by 𝑥 plus three plus two multiplied by 𝑥 plus three is equal to 𝑥 plus two multiplied by 𝑥 plus three. This expression cannot be factored any further as the two terms in each binomial do not share any common factors other than one.

We now consider how to factor a quadratic expression of the form 𝑥 squared plus 𝑏𝑥 plus 𝑐 into the product of two binomials. Factoring is the reverse process of distributing parentheses or expanding brackets. Consider the expansion of the product of the binomials 𝑥 plus five and 𝑥 plus three. We begin by distributing the first set of parentheses over the second, giving us 𝑥 multiplied by 𝑥 plus five plus three multiplied by 𝑥 plus five. Distributing each set of parentheses, we have 𝑥 squared plus five 𝑥 plus three 𝑥 plus 15. And collecting like terms, this simplifies to 𝑥 squared plus eight 𝑥 plus 15.

We can observe that the constant term in the trinomial expression is the product of a constant terms in the two binomials. 15 is equal to three multiplied by five. The coefficient of 𝑥 in the trinomial is the sum of the constant terms in the two binomials. Eight is equal to three plus five. And in the penultimate line of our solution, the 𝑥-term is written as the sum of two terms with these coefficients, five 𝑥 plus three 𝑥. This suggests a process we can follow to work in the opposite direction and factor an expanded quadratic of the form 𝑥 squared plus 𝑏𝑥 plus 𝑐 into the product of two binomials. It is important to note that not all quadratics of this form can be factored, so the process that follows is only applicable to those that are factorable.

Firstly, we list the factor pairs of the constant 𝑐. If 𝑐 is positive, the two numbers will have the same sign, whereas if 𝑐 is negative, the two numbers will have opposite signs. Look for a factor pair that with the correct combinations of signs sum to the coefficient of 𝑥, i.e., 𝑏. Rewrite the middle term in the trinomial as the sum of two terms with coefficients equal to the factor pair found. Separate the new four-term expression into two binomials and factor each. Look for a shared binomial factor to factor the entire expression by. We will demonstrate this process of factoring a quadratic of the form 𝑥 squared plus 𝑏𝑥 plus 𝑐 in our next example.

Factor 𝑥 squared minus eight 𝑥 minus 20.

To factor this quadratic expression, we wish to write it as the product of two binomials. We will do this by first rewriting the middle term as the sum of two terms with coefficients whose sum is the coefficient of 𝑥 and whose product is the constant term. We first consider the factor pairs of 20. As the product of the two numbers must be negative 20, the two numbers must have different signs. If we choose the second factor pair of two and 10 and choose the two to be positive and the 10 to be negative, then the sum of these two numbers is two plus negative 10, which is equal to negative eight as required.

We then rewrite the trinomial with the middle term expressed as the sum of two terms with coefficients of two and negative 10, that is, 𝑥 squared plus two 𝑥 minus 10 𝑥 minus 20. Separating this four-term expression into two binomials and factoring gives us 𝑥 multiplied by 𝑥 plus two minus 10 multiplied by 𝑥 plus two. Finally, we factor the entire expression by the shared binomial factor of 𝑥 plus two to give 𝑥 plus two multiplied by 𝑥 minus 10. This is the fully factored form of 𝑥 squared minus eight 𝑥 minus 20.

We have seen in this example how to factor quadratics of the form 𝑥 squared plus 𝑏𝑥 plus 𝑐 into the product of two binomials. Such quadratics, in which the coefficient of 𝑥 squared is equal to one, are known as monic quadratics. We will now consider the more general case of how to factor a nonmonic quadratic of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where 𝑎 is not equal to zero, one, or negative one. We will demonstrate this process through an example.

Consider the product of the binomials three 𝑥 minus two and two 𝑥 plus five.

Using the distributive property, we have three 𝑥 multiplied by two 𝑥 plus five minus two multiplied by two 𝑥 plus five, which simplifies to six 𝑥 squared plus 15𝑥 minus four 𝑥 minus 10 and in turn to six 𝑥 squared plus 11𝑥 minus 10. Now we consider the reverse of this, expressing the trinomial six 𝑥 squared plus 11𝑥 minus 10 as the product of two binomials. We observe that in the first line of our working, the two parts of the expression had a shared binomial factor of two 𝑥 plus five.

To work in the other direction, we first need to rewrite a trinomial as an expression involving four terms so that we can then separate the resulting expression into one that contains two binomials and factor each separately. We begin by looking for two numbers whose sum is the coefficient of 𝑥, in this example 11, and whose product is equal to the product of the coefficient of 𝑥 squared and the constant term, in this case six multiplied by negative 10, which is equal to negative 60. These two numbers are 15 and negative four. We then rewrite the trinomial as a four-term expression, separating the 𝑥-term into the sum of two terms with these coefficients.

Next, we split this expression into two binomials and factor each binomial separately, giving us three 𝑥 multiplied by two 𝑥 plus five minus two multiplied by two 𝑥 plus five. This reveals a common binomial factor of two 𝑥 plus five, which can subsequently be factored to give our solution two 𝑥 plus five multiplied by three 𝑥 minus two. This method of factoring is the reverse process of expanding the brackets that we see on the right-hand side of the screen. The method for factoring a monic quadratic that we met earlier is, in fact, a special case of this method, in which the product of 𝑎 and 𝑐 is equal to 𝑐 because 𝑎 equals one.

The examples we have considered so far have concerned trinomials involving only a single variable. In our final example, we will consider how to factor a trinomial involving two variables.

Factorize fully 48𝑚 to the fourth power plus 48𝑚 squared 𝑛 minus 15𝑛 squared.

We begin by observing that the coefficients of all three terms are multiples of three. Hence, we can factor the entire trinomial by three, which gives us three multiplied by 16𝑚 to the fourth power plus 16𝑚 squared 𝑛 minus five 𝑛 squared. Now consider the structure of the trinomial. The first term involves 𝑚 to the fourth power, which is equal to 𝑚 squared all squared. And the third term involves 𝑛 squared. The central term involves a product of 𝑚 squared and 𝑛. This suggests that the factored form of the trinomial is 𝐴𝑚 squared plus 𝐵𝑛 multiplied by 𝐶𝑚 squared plus 𝐷𝑛, for values of 𝐴, 𝐵, 𝐶, and 𝐷 to be determined.

We now need to find two numbers whose sum is equal to the coefficient of 𝑚 squared 𝑛, in this case 16, and whose product is equal to the product of the coefficients of the first and last terms. 16 multiplied by negative five is equal to negative 80. The factor pairs of 80 are as shown. As the product should be negative 80, we require a factor pair with opposite signs such that their sum is equal to 16. The correct pair are 20 and negative four. Rewriting the second term in the trinomial as the sum of two terms with these coefficients gives 16𝑚 to the fourth power plus 20𝑚 squared 𝑛 minus four 𝑚 squared 𝑛 minus five 𝑛 squared.

Separating this four-term expression into two binomials and factoring each separately gives four 𝑚 squared multiplied by four 𝑚 squared plus five 𝑛 minus 𝑛 multiplied by four 𝑚 squared plus five 𝑛. Hence, the fully factored form of the trinomial is three multiplied by four 𝑚 squared minus 𝑛 multiplied by four 𝑚 squared plus five 𝑛.

Let us finish by recapping the key points from this video.

To factor a nonmonic quadratic of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where 𝑎 is not equal to zero, one, or negative one, we perform the following steps. List the factor pairs of 𝑎𝑐. Look for a factor pair that with the correct combination of signs sum to the coefficient of 𝑥, noting that if 𝑎𝑐 is positive, the two numbers will have the same sign, whereas if 𝑎𝑐 is negative, the two numbers will have opposite signs. Rewrite the middle term in the trinomial as the sum of terms with coefficients equal to the factor pair found. Separate the new four-term expression into two binomials and factor each. Look for a shared binomial factor to factor the entire expression by.

Factoring monic quadratics of the form 𝑥 squared plus 𝑏𝑥 plus 𝑐 is a special case of the above, where 𝑎 is equal to one. And hence, 𝑎𝑐 is equal to 𝑐. To factor a two-variable trinomial, first consider its structure and identify the structure of the binomial factors. The coefficients can be found using the same method used for nonmonic quadratics. Note that some trinomials can be factored into the product of more than two terms by first taking out a common factor.

Whilst we will not cover it in this video, the techniques we have encountered here can be applied to problems in other areas of mathematics, such as geometry or real-world problems.