Lesson Video: Factoring Monic Quadratics Mathematics

Video Transcript

In this video, we will learn how to factor quadratic expressions in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where the coefficient of the leading term is one. We will begin by defining what we mean by a quadratic expression.

A quadratic trinomial is an algebraic expression in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐. 𝑎 is the leading or quadratic coefficient, 𝑏 is the linear coefficient, and 𝑐 is the constant term. A monic quadratic trinomial, on the other hand, is an expression in the form 𝑥 squared plus 𝑏𝑥 plus 𝑐. The leading or quadratic coefficient is equal to one. In this video, we will look at quadratics of this type that can be factored into two sets of parentheses or brackets. We will rewrite 𝑥 squared plus 𝑏𝑥 plus 𝑐 in the form 𝑥 plus 𝑝 multiplied by 𝑥 plus 𝑞.

If we recall our FOIL method for distributing two sets of parentheses, multiplying the first terms gives us 𝑥 squared. Multiplying the outside or outer terms gives us 𝑞𝑥. Multiplying the inner terms gives us 𝑝𝑥. Finally, multiplying the last terms gives us 𝑝𝑞. We can simplify the middle two terms by factoring out an 𝑥. 𝑞𝑥 plus 𝑝𝑥 is equal to 𝑝 plus 𝑞 multiplied by 𝑥. This means that the expansion of 𝑥 plus 𝑝 multiplied by 𝑥 plus 𝑞 is equal to 𝑥 squared plus 𝑝 plus 𝑞 multiplied by 𝑥 plus 𝑝𝑞.

When comparing this with 𝑥 squared plus 𝑏𝑥 plus 𝑐, the 𝑏 has been replaced by 𝑝 plus 𝑞. This means that the linear coefficient 𝑏 must be equal to 𝑝 plus 𝑞. In the same way, the constant term 𝑐 must be equal to 𝑝 multiplied by 𝑞. We can therefore conclude that when factoring a monic quadratic, the sum of the numbers 𝑝 and 𝑞 must equal 𝑏 and the product of 𝑝 and 𝑞 must equal 𝑐.

We will now look at some questions where we need to factor a monic quadratic.

Factor 𝑥 squared plus eight 𝑥 plus 12.

In this question, we have a quadratic of the form 𝑥 squared plus 𝑏𝑥 plus 𝑐. As the leading coefficient or quadratic coefficient is equal to one, this can be factored into two sets of parentheses, 𝑥 plus 𝑝 multiplied by 𝑥 plus 𝑞. For any quadratic of this type, 𝑏 is equal to 𝑝 plus 𝑞 and 𝑐 is equal to 𝑝 multiplied by 𝑞. We are looking for two numbers that have a product of 12 and a sum of eight. The number 12 has three sets of factor pairs: one and 12, two and six, and three and four. Of these three pairs, only two and six also sum to eight. The expression 𝑥 squared plus eight 𝑥 plus 12 in its factored form is equal to 𝑥 plus two multiplied by 𝑥 plus six.

We can check this answer by expanding the brackets or distributing the parentheses. One way of doing this is using the FOIL method. Multiplying the first terms gives us 𝑥 squared. Multiplying the outer terms gives us six 𝑥. Multiplying the inner terms gives us two 𝑥. And finally, multiplying the last terms gives us 12. Collecting like terms, this simplifies to 𝑥 squared plus eight 𝑥 plus 12. As this is the same as our original expression, we know that our answer 𝑥 plus two multiplied by 𝑥 plus six is correct.

In this question, both 𝑏 and 𝑐 were positive. In our next example, we will look at a case where 𝑏 and 𝑐 are negative.

Factor 𝑥 squared minus eight 𝑥 minus 20.

This quadratic expression has a leading coefficient of one. This means it is of the type 𝑥 squared plus 𝑏𝑥 plus 𝑐. If this quadratic can be factored, we can write it as the product of two sets of parentheses, 𝑥 plus 𝑝 multiplied by 𝑥 plus 𝑞. The sum of 𝑝 plus 𝑞 is equal to the coefficient of 𝑥, 𝑏. The product of 𝑝 and 𝑞 is equal to the constant term 𝑐.

We need to find two numbers that have a product of negative 20 and a sum of negative eight. Let’s begin by considering the factor pairs of 20. They are one and 20, two and 10, and four and five. As we want the product of these numbers to be negative, one must be positive and one must be negative. There is no way that the pairs one and 20 and four and five can lead us to a sum of negative eight. If we had negative two and positive 10, these would have a sum of positive eight. This means that we actually want positive two and negative 10. These two numbers have a product of negative 20 and a sum of negative eight. This means that the factored form of 𝑥 squared minus eight 𝑥 minus 20 is 𝑥 plus two multiplied by 𝑥 minus 10.

We could check this answer by distributing our parentheses. Multiplying 𝑥 by 𝑥 gives us 𝑥 squared. Multiplying 𝑥 by negative 10 gives us negative 10𝑥. Two multiplied by 𝑥 is equal to two 𝑥. And two multiplied by negative 10 is negative 20. As this simplifies to 𝑥 squared minus eight 𝑥 minus 20, which is the same as our original expression, we know that our answer is correct.

We will now look at a more complicated problem where we need to expand or distribute our parentheses first.

Expand and simplify 𝑎 multiplied by 𝑎 plus nine plus 18, then factorize the result.

In order to expand the brackets or distribute the parentheses, we need to multiply 𝑎 by 𝑎 and 𝑎 by nine. This gives us 𝑎 squared plus nine 𝑎. We can drop down the constant term 18 so that 𝑎 multiplied by 𝑎 plus nine plus 18 is equal to 𝑎 squared plus nine 𝑎 plus 18. This is a quadratic expression written in the form 𝑥 squared plus 𝑏𝑥 plus 𝑐. The leading coefficient of the quadratic is one. This means that it can be factored into two sets of parentheses, 𝑥 plus 𝑝 multiplied by 𝑥 plus 𝑞.

The first term in both of our parentheses will be 𝑎 as 𝑎 multiplied by 𝑎 is 𝑎 squared. The numbers 𝑝 and 𝑞 must sum to give us 𝑏, and they must multiply or have a product of 𝑐. We need to find two numbers that have a sum of nine and a product of 18. There are three pairs of integers that multiply to give us 18: one and 18, two and nine, and three and six. The only one of these pairs that has a sum of nine is three and six. 𝑎 squared plus nine 𝑎 plus 18 is equal to 𝑎 plus three multiplied by 𝑎 plus six. This means that our initial expression 𝑎 multiplied by 𝑎 plus nine plus 18 is also equal to 𝑎 plus three multiplied by 𝑎 plus six.

Our next example is a bit of a trick question as we actually start with a cubic equation.

Factorize fully 𝑥 cubed plus two 𝑥 squared minus 63𝑥.

This expression is a cubic as the highest exponent is a three. This suggests that it could be quite difficult to factor or factorize. However, when we look at the expression more closely, we notice that all three terms have a common factor of 𝑥. This means that we can take this factor outside our parentheses. 𝑥 cubed divided by 𝑥 is equal to 𝑥 squared. Two 𝑥 squared divided by 𝑥 is equal to two 𝑥. Finally, negative 63𝑥 divided by 𝑥 is negative 63.

We now have a quadratic expression of the form 𝑥 squared plus 𝑏𝑥 plus 𝑐. As the coefficient of 𝑥 squared is equal to one, we can try and factor this into two sets of parentheses, 𝑥 plus 𝑝 multiplied by 𝑥 plus 𝑞. The sum of 𝑝 and 𝑞 will be equal to 𝑏, and the product of 𝑝 and 𝑞 will be equal to 𝑐. We need to try and find two integers with a product of negative 63 and a sum of two. We know that nine multiplied by seven is equal to 63. Multiplying a positive number by a negative number gives a negative answer. Therefore, positive nine multiplied by negative seven or negative nine multiplied by positive seven both give us negative 63. As we want our numbers to have a sum of positive two, the correct pair is positive nine and negative seven.

The expression 𝑥 squared plus two 𝑥 minus 63 is equal to 𝑥 plus nine multiplied by 𝑥 minus seven. This means that the fully factorized form of 𝑥 cubed plus two 𝑥 squared minus 63𝑥 is equal to 𝑥 multiplied by 𝑥 plus nine multiplied by 𝑥 minus seven. We could expand this expression by distributing our parentheses to check that our answer is correct.

In our next question, we need to work out the missing term in a quadratic expression.

Given that the expression 𝑥 squared plus 𝑎𝑥 minus 35 can be factorized, what is the set of all possible integer values of 𝑎?

The expression in this question is written in the form 𝑥 squared plus 𝑏𝑥 plus 𝑐. The leading coefficient or coefficient of 𝑥 squared is equal to one. We are told that it can be factored or factorized. This means we can write it in the form 𝑥 plus 𝑝 multiplied by 𝑥 plus 𝑞. The coefficient of 𝑥, the value 𝑏, will be equal to 𝑝 plus 𝑞. And the constant term 𝑐 will be equal to 𝑝 multiplied by 𝑞. This means that in our expression, the two values 𝑝 and 𝑞 must have a sum of 𝑎 and a product of negative 35. The number 35 has two factor pairs: one and 35 and five and seven.

As multiplying a negative number by a positive number gives a negative answer, there are four pairs of integers that multiply to give us negative 35. We have negative one and positive 35, positive one and negative 35, negative five and positive seven, and five and negative seven. These are the potential values of 𝑝 and 𝑞. We need to find the sum of each of these pairs of values as we know that the coefficient of 𝑥 will be equal to 𝑝 plus 𝑞. Negative one plus 35 is equal to 34. The sum of one and negative 35 is negative 34. Negative five plus seven is equal to two. And five plus negative seven is equal to negative two. This means that the set of all possible integer values of 𝑎 in ascending order are negative 34, negative two, two, and 34.

These values occur when our sets of parentheses are 𝑥 minus 35 multiplied by 𝑥 plus one, 𝑥 minus seven multiplied by 𝑥 plus five, 𝑥 minus five multiplied by 𝑥 plus seven, and 𝑥 minus one multiplied by 𝑥 plus 35, respectively. After expanding these brackets, the 𝑥-term would become negative 34𝑥, negative two 𝑥, two 𝑥, and 34𝑥, giving our values of 𝑎 of negative 34, negative two, two, and 34.

We will now summarize the key points from this video. A monic quadratic expression has a leading coefficient of one. This means it can be written in the form 𝑥 squared plus 𝑏𝑥 plus 𝑐. If an expression of this form can be factored, it can be written 𝑥 plus 𝑝 multiplied by 𝑥 plus 𝑞. The coefficient of 𝑥 is equal to 𝑝 plus 𝑞, and the constant term is equal to 𝑝 multiplied by 𝑞. This means that in order to factor an expression of this type, we need to find two integers that have a product equal to 𝑐 and a sum equal to 𝑏. Using this information, we also saw how we could find the missing term in a quadratic expression.