Lesson Video: Multiplying Polynomials Mathematics

Video Transcript

In this video, we will learn how to multiply polynomial expressions together by expanding the parentheses.

To begin, let’s review definitions of a few terms that we will be using in this lesson. First, a monomial is a product of constants and variables, where the variables may contain only nonnegative integer exponents. Second, a polynomial is an expression that is a sum of single monomial terms. We recall that a polynomial with two terms is called a binomial. For example, eight 𝑥 cubed minus one is a binomial. We also recall that a polynomial with three terms is called a trinomial, such as 𝑥 squared minus four 𝑥 plus four.

In order to multiply any two polynomials together, we should first recall how we multiply a polynomial by a monomial. To do this, we distribute the multiplication over each term in the polynomial. For example, let’s multiply 𝑥 times the polynomial 𝑥 minus 𝑦. First, we distribute 𝑥 over the 𝑥-term then the 𝑦-term, resulting in 𝑥 times 𝑥 minus 𝑥 times 𝑦. Then, after expanding this product, we simplify the expression to 𝑥 squared minus 𝑥𝑦.

We can apply this exact same process to multiply any two polynomials. Let’s consider the product 𝑥 plus one times 𝑥 plus two. This is not quite the same form since neither factor is a monomial. However, we can treat the factor 𝑥 plus one as a single term. This means we just multiply each term of 𝑥 plus two by the entire factor 𝑥 plus one as follows. Each term is now the product of a monomial and binomial. So, we can expand the expression by distributing 𝑥 over 𝑥 plus one and two over 𝑥 plus one. So, we have 𝑥 times 𝑥 plus 𝑥 times one plus two times 𝑥 plus two times one. Then, we simplify each product and finally collect the like terms. In particular, we add the linear terms by combining their coefficients. So, we have shown that 𝑥 plus one times 𝑥 plus two is equal to 𝑥 squared plus three 𝑥 plus two.

In general, we can apply this process to find the product of any two polynomials. We distribute one factor over every term in the second factor, expand, and then simplify. Let’s see an example of applying this process to find the product of two binomials.

Expand two 𝑥 plus five squared.

We’re asked to expand the square of a binomial. We can do this by first recalling that squaring an expression means multiplying that expression by itself. So, two 𝑥 plus five squared is equal to two 𝑥 plus five times two 𝑥 plus five. We can now expand this product by distributing the first factor over every term in the second factor. So, we multiply two 𝑥 by two 𝑥 plus five and five by two 𝑥 plus five. This is written as two 𝑥 times two 𝑥 plus five plus five times two 𝑥 plus five. Then, we expand each term by distributing the factor over each binomial, like so: two 𝑥 times two 𝑥 plus two 𝑥 times five plus five times two 𝑥 plus five times five.

We can now simplify each term by recalling that 𝑥 times 𝑥 is equal to 𝑥 squared and collecting like terms. So we have four 𝑥 squared plus 10𝑥 plus 10𝑥 plus 25, which simplifies to four 𝑥 squared plus 20𝑥 plus 25. This is the square of two 𝑥 plus five.

We can also answer this question by recalling that we can square a binomial using the formula 𝑎 plus 𝑏 squared equals 𝑎 squared plus two 𝑎𝑏 plus 𝑏 squared. Substituting 𝑎 equals two 𝑥 and 𝑏 equals five into the formula yields two 𝑥 squared plus two times two 𝑥 times five plus five squared, which simplifies to four 𝑥 squared plus 20𝑥 plus 25.

In our next example, we will expand and simplify the product of a trinomial and a binomial with multiple variables and a negative coefficient.

Expand and simplify 𝑥 minus two 𝑦 plus three times two 𝑥 plus 𝑦.

To expand the product of two polynomials, we need to distribute one polynomial over each term in the other. So that means we could distribute the trinomial over the binomial or the binomial over the trinomial. In this case, it seems the first option is more straightforward. So, let’s start with distributing the trinomial over each term in the binomial as follows. We can then expand each of the two terms by distributing the first factor over each trinomial. For the first term, we have two 𝑥 times 𝑥 plus two 𝑥 times negative two 𝑦 plus two 𝑥 times three, which simplifies to two 𝑥 squared minus four 𝑥𝑦 plus six 𝑥. For the second term, we have 𝑦 times 𝑥 plus 𝑦 times negative two 𝑦 plus 𝑦 times three, which simplifies to 𝑥𝑦 minus two 𝑦 squared plus three 𝑦.

Now we can add the expressions together and collect like terms to simplify. We obtain two 𝑥 squared minus three 𝑥𝑦 minus two 𝑦 squared plus six 𝑥 plus three 𝑦. We have fully expanded and simplified the product of 𝑥 minus two 𝑦 plus three times two 𝑥 plus 𝑦.

In our next example, we will use this process of multiplying polynomials to find a simplified expression for a shaded region.

A rectangle with dimensions 𝑥 minus 𝑦 and 𝑥 plus 𝑦 plus one is cut out from a larger rectangle with dimensions two 𝑥 plus 𝑦 plus three and two 𝑥 plus 𝑦. Find a simplified expression for the shaded area.

We can find an expression for the shaded area by finding the area of the larger rectangle and subtracting the area of the smaller rectangle from the area of the larger rectangle. We recall the area of a rectangle is found by multiplying its length times its width. Let’s start by expanding and simplifying the expression for the area of the larger rectangle, two 𝑥 plus 𝑦 plus three times two 𝑥 plus 𝑦. We can do this by distributing the trinomial over each term in the binomial. So, we have two 𝑥 times two 𝑥 plus 𝑦 plus three plus 𝑦 times two 𝑥 plus 𝑦 plus three.

Then, we can distribute the monomials over the trinomials. Distributing two 𝑥 over two 𝑥 plus 𝑦 plus three gives four 𝑥 squared plus two 𝑥𝑦 plus six 𝑥. And distributing 𝑦 over the same trinomial gives two 𝑥𝑦 plus 𝑦 squared plus three 𝑦. We can add these expressions together and collect like terms to find the area of the larger rectangle, four 𝑥 squared plus four 𝑥𝑦 plus 𝑦 squared plus six 𝑥 plus three 𝑦.

Now, let’s clear some space to find the area of the smaller rectangle. For the smaller rectangle, we have dimensions 𝑥 plus 𝑦 plus one and 𝑥 minus 𝑦. So, the expression for its area comes from the product of these two expressions. We can do this, once again, by distributing the trinomial over each term in the binomial. So, we have 𝑥 times 𝑥 plus 𝑦 plus one minus 𝑦 times 𝑥 plus 𝑦 plus one. Then, we can distribute the monomials over the trinomials. Distributing 𝑥 over 𝑥 plus 𝑦 plus one gives 𝑥 squared plus 𝑥𝑦 plus 𝑥. And distributing negative 𝑦 over 𝑥 plus 𝑦 plus one gives negative 𝑥𝑦 minus 𝑦 squared minus 𝑦. We can then add these expressions together and collect like terms to find the area of the smaller rectangle, resulting in the expression 𝑥 squared minus 𝑦 squared plus 𝑥 minus 𝑦.

Now we need to subtract the area of the smaller rectangle from the area of the larger rectangle to find the area of the shaded region. We must be careful on this step to subtract all the terms of the second polynomial, not just the first term. We can do this by distributing the negative through each term and then collecting the like terms. Thus, we subtract 𝑥 squared, add 𝑦 squared, subtract 𝑥, and add 𝑦 to the first expression. We can use a vertical method to stack the like terms, resulting in three 𝑥 squared plus four 𝑥𝑦 plus two 𝑦 squared plus five 𝑥 plus four 𝑦. This is the simplified expression for the shaded region, which we found by subtracting the area of the smaller rectangle from the area of the larger rectangle.

In our final example, we will expand the product of three binomials to find the values of unknown coefficients.

If two 𝑥 minus 𝑦 times two 𝑥 minus five 𝑦 times three 𝑥 plus two 𝑦 is equal to 𝑎𝑥 cubed plus 𝑏𝑥 squared 𝑦 plus 𝑐𝑥𝑦 squared plus 𝑑𝑦 cubed, what are the values of 𝑎, 𝑏, 𝑐, and 𝑑?

On the left-hand side of the equation, we have the product of three binomials. And on the right-hand side of the equation, we have a polynomial. We can find a polynomial expression equivalent to the left-hand side of the equation by expanding the product. Let’s start off by finding the product of the first two factors. To do this, we can take the product of each pair of terms from the first parentheses with the second parentheses and add the results.

So, we can distribute the first term from the first binomial through both terms in the second binomial, resulting in two 𝑥 times two 𝑥 plus two 𝑥 times negative five 𝑦. Then, we can distribute the second term from the first binomial through both terms in the second binomial, resulting in negative 𝑦 times two 𝑥 plus negative 𝑦 times negative five 𝑦. Simplifying the results gives four 𝑥 squared minus 10𝑥𝑦 minus two 𝑥𝑦 plus five 𝑦 squared. Then, we can combine like terms to get four 𝑥 squared minus 12𝑥𝑦 plus five 𝑦 squared. Now we can substitute this expression into our product of three binomials. Then, we can expand the product by distributing the trinomial over each term in the binomial.

For the first term, we have 12𝑥 cubed minus 36𝑥 squared 𝑦 plus 15𝑥𝑦 squared. And for the second term, we have eight 𝑥 squared 𝑦 minus 24𝑥𝑦 squared plus 10𝑦 cubed. Combining the like terms and simplifying gives 12𝑥 cubed minus 28𝑥 squared 𝑦 minus nine 𝑥𝑦 squared plus 10𝑦 cubed. We are told this is equal to 𝑎𝑥 cubed plus 𝑏𝑥 squared 𝑦 plus 𝑐𝑥𝑦 squared plus 𝑑𝑦 cubed. For the polynomials to be equal, their coefficients must be equal. Thus, 𝑎 equals 12, 𝑏 equals negative 28, 𝑐 equals negative nine, and 𝑑 equals 10.

Let’s finish by recapping some of the important points from this video. We can multiply two polynomials by first multiplying every term of one polynomial by the other entire polynomial then summing their results. We can finally simplify by combining like terms. We also saw in our last example that the product of any two polynomials can also be found by multiplying every pair of terms from each of the polynomials and summing the results. So for two binomials 𝑎 plus 𝑏 and 𝑐 plus 𝑑, this means their product can be found by summing 𝑎 times 𝑐 plus 𝑎 times 𝑑 plus 𝑏 times 𝑐 plus 𝑏 times 𝑑.