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The degree of a polynomial is guaranteed to be its largest exponent if it contains only one variable. If the polynomial in question contains more than one variable, its degree is the highest degree out of all the polynomial's terms, which is not necessarily the polynomial's greatest exponent. The highest degree is the highest power. This is easiest to figure out if the polynomial is in standard form.

For example, the degree of $x^2 + 3$ is $2$ because the highest exponent of the variable $x$ is $2$ (in the $x^2$ term).

The degree of $x^3y^3 + x^2y^2 + 5$ is $6$ (not $3$) because the highest degree of any of its terms is $6:$

The $5$ term in the polynomial above does not have any variables in it. Thus, the degree is $0.$

The $x^2y^2$ term has an $x$ raised to the $2$ and a $y$ raised to the $2.$ Thus, the degree of this term is $2+2=4.$

The $x^3y^3$ term has an $x$ raised to the $3$ and a $y$ raised to the $3.$ Thus, the degree of this term is $3+3=6.$

The term with the highest degree has degree $6,$ so $6$ is the degree of the whole polynomial.

Scalar Multiplication

The degree of the product of a polynomial and a non-zero real number is the degree of the polynomial itself.

For example, the degree of the product of $a$ (some non-zero real number) and $x^3 + x^2 + 1$ is $3.$ Because $a\cdot (x^3 + x^2 + 1) = ax^3 + ax^2 + a.$ The highest degree of this polynomial is equal to the biggest exponent, which is $3.$

Multiplication of Two or More Polynomials

The degree of the product of two or more polynomials with one variable is the sum of the degrees of each polynomial.

For example, the degree of the product of $x^2 + 1$ and $4x^3 + 5x + 1$ is $5$. This is because the degree of $x^2 + 1$ is $2,$ and the degree of $4x^3 + 5x + 1$ is $3,$ so the total degree is $2+3=5.$

The degree of a term in the product of two or more polynomials with two or more variables is the sum of the exponents of each variable in the term. The degree of the polynomial as a whole is equal to the highest degree of any of the terms.

For example, consider $xy + x^2y.$

The term $xy$ has degree $2,$ because $x$ is raised to the $1$ and $y$ is raised to the $1,$ and $1+1=2.$

The term $x^2y$ has degree $3,$ because $x$ is raised to the $2$ and $y$ is raised to the $1,$ and $2+1=3.$

The polynomial $xy + x^2y$ has a degree of $3$ because $3$ is the highest degree of either of its terms!

The degree of a product (which will be a polynomial, itself) of two or more polynomials with multiple variables is equal to the sum of the degrees of each polynomial.

For example, consider the product of $xy + x^2y$ and $xy^2 + y^3 + x$.

The degree of $xy + x^2y$ is $3,$ as we have seen in the previous example. The degree of $xy^2 + y^3 + x$ is $3.$ (This is because the degree of the term with the highest degree, $xy^2,$ is $3$.) Thus, the degree of the product of these two polynomials is $3+3=6.$

When we multiply those two polynomials together and simplify the resulting expression, we get $x^2y^3 + xy^4 + x^2y + x^3y^3 + x^2y^4 + x^3y$. Note that there are two terms in this product that are tied for highest degree: $x^3y^3$ and $x^2y^4$. Both of these terms have a degree of $6,$ which means that the degree of the entire product is $6.$

Test Yourself!

Consider the following expression: $(a^2b^2 + a)(a^4 + b + a)(b^3 + a)$

Note: the degree of a polynomial must be a non-negative integer.

When we multiply two terms with the same base such as $x^{2}$ and $x^{3}$, our product is the base raised to the sum of the exponents, like so: $$x^{2+3} = x^{5}$$

This is the highest power.

Now, let's see how we can apply this knowledge to products of polynomials.

Consider two generic polynomials of degree $n$ and $m$: $$a_n x^{n} + a_{n-1} x^{n-1}... + a_0$$ $$b_m x^{m} + b_{m-1} x^{m-1}... + b_0$$ Let's think about how we go about finding the degree of the product of those two polynomials. Well, when we multiply polynomials, their terms are all multiplied out. With that in mind, consider the product that would result from multiplying the two polynomials above: which of its terms has the largest exponent (and therefore determines the product's degree)?

In order to find the greatest exponent of our product, we need to find the greatest exponents of our two polynomials, which are $m$ and $n$, respectively. Therefore, the degree of our product is $m+n$.

Now, you may think, what would happen if I multiplied many polynomials together. For the sake of ease, let's just multiply four polynomials together, with degrees $k,l,m$ and $n$. Multiplying the first two polynomials results in a polynomial of degree $k+l$. We then multiply this new polynomial by the polynomial of degree $m$, and then multiply that product by the polynomial of degree $n$. In the end, this results in a polynomial of degree $k+l+n+m$. Note that multiplication of polynomials is commutative and associative, which is convenient.

Another way to look at this is through the lens of the fundamental theorem of algebra (FDA). Again, we refer back to our two polynomials of degree $m$ and $n$. By the FDA, those polynomials have exactly $m$ and $n$ roots, respectively. Effectively, we've factored our polynomials. Multiplication of the polynomials is synonymous to multiplying their factored versions, which results in a polynomial with $m+n$ roots. Reversing what we previously did, we recognize that this product must have a degree of $m+n$.

(Video) Find the Degree of a Polynomial Product without Multiplying

by Anusha Rahman

In this video by Anusha Rahman, you will learn how to find the degree of a polynomial product without multiplying out the polynomial.

Summary

When you multiply two polynomials together, the degree of the resulting polynomial is the sum of the degrees of the individual polynomials.

Given $f(x)$ with degree $m$ and $g(x)$ with degree $n$,

$$h(x) = f(x) \cdot g(x)$$

The degree of $h(x)$ is $m+n$

Finding The Degree of a Polynomial Product

Now that we've covered the basics of polynomial operations, let's move into the degree. The degree of a polynomial determines both its end behavior and overall shape.

In this lesson, we will learn how to find the degree of a polynomial product. First, remember that the degree of a polynomial is the highest exponent of the variable (usually $x$) in that polynomial when it is written in standard form.

When we multiply two polynomials together, the degree of the resulting polynomial is the sum of the degrees of the polynomials being multiplied. For example, if you multiply a polynomial with degree $2$ by a polynomial with degree $3$, the degree of the product will be $2 + 3 = 5$.

Let's look at an example.

Example

Suppose you have the polynomials $(2x^2 + 3x+1)$ and $(x^3+4x-2)$. The degree of the first polynomial is $2$, and the degree of the second polynomial is $3$. When you multiply them, the degree of the product will be $2+3=5$.

Let's think about the multiplication of the highest degree terms: $(2x^2) \cdot (x^3) = 2x^5$. The highest degree term is $2x^5$, which confirms that the degree of the product is $5$.

It's important to note that you do not need to actually multiply out every term to find the degree. Just focus on the highest degree terms in each polynomial. The product's degree is simply the sum of the degrees of these highest degree terms.

Try this next example on your own!

Example

What is the degree of the polynomial product?

$(x^4 + 2x^2) \cdot (x^3-x+5)$

The degree of the product will be $4+3=7$.

Remember, when multiplying polynomials, always add the degrees of the highest-degree terms to find the degree of the product. Keep practicing, and you'll get the hang of it!