Introduction to Polynomials

What Are Polynomials?

A polynomial is an expression containing constants and variables connected only through basic operations of algebra.

Learning Objectives

Describe what polynomials are and their defining characteristics

Key Takeaways

Key Points

  • A polynomial is a finite expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and taking non-negative integer powers.
  • A polynomial can be written as the sum of a finite number of terms. Each term consists of the product of a constant (called the coefficient of the term) and a finite number of variables (usually represented by letters) raised to integer powers.

Key Terms

  • polynomial: an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as [latex]a_n x^n + a_{n-1}x^{n-1} +… + a_0 x^0[/latex].
  • degree: the sum of the exponents of a term; the order of a polynomial.
  • coefficient: a constant by which an algebraic term is multiplied.

Polynomials are widely used algebraic objects. They have the form of a sum of scaled powers of a variable.

Monomials over [latex]\mathbb{R}[/latex]

Let [latex]\mathbb{R}[/latex] be the set of real numbers. A monomial over [latex]\mathbb{R}[/latex] in a single variable [latex]x[/latex] consists of a non-negative power of [latex]x[/latex], multiplied with a nonzero constant [latex]c \in \mathbb{R}.[/latex] So a polynomial looks like

[latex]cx^n[/latex],

where [latex]n \geq 0[/latex] is an integer and [latex]c \not = 0[/latex] is a real number. If we want to give the polynomial a name, say [latex]M[/latex], we denote that its variable is [latex]x[/latex] by writing [latex]x[/latex] between brackets:

[latex]M(x)=cx^n[/latex].

The exponent [latex]n[/latex] is called the degree of [latex]M(x).[/latex] The constant [latex]c[/latex] is the coefficient.

Examples

[latex]\sqrt{2}x^7[/latex]is a monomial of degree 7 and coefficient [latex]\sqrt{2}[/latex].

[latex]7x^{\sqrt{2}}[/latex], [latex]\sqrt{2}x^{-7}[/latex] and [latex]2x^7 - 7x^2[/latex]are not monomials. The first and the second do not have a non-negative integer exponent and the third is a sum of two monomials.

Polynomials over [latex]\mathbb{R}[/latex]

A polynomial over [latex]\mathbb{R}[/latex] is a finite sum of monomials over [latex]\mathbb{R}[/latex]. For example

[latex]P(x)= 4x^{13} +3x^2-\pi x + 1[/latex]

is the finite sum of the [latex]4[/latex] monomials: [latex]4x^{13}, 3x^2, -\pi x[/latex] and [latex]1 = 1x^0.[/latex]

It is also the sum of the 6 monomials: [latex]1/3 x^{100}, -1/3 x^{100}, 4x^{13}, 3x^2, -\pi x[/latex] and [latex]1[/latex], as will be explained in the discussion about addition and subtraction of polynomials. However, we can only write down [latex]P(x)[/latex] as the sum of monomials of distinct degree in exactly one way, namely the first we mentioned. These monomials are called the terms of [latex]P(x).[/latex]The coefficients of [latex]P(x)[/latex] are the coefficients corresponding to its terms.

Every monomial is also a polynomial, as it can be written as a sum with one term, itself.

A special example of a polynomial is the zero polynomial

[latex]Z(x) = 0,[/latex]

which is a sum of [latex]0[/latex] monomials.

The degree of a polynomial [latex]Q(x)[/latex] is the highest degree of one of its terms. For example, the degree of [latex]P(x)[/latex] is [latex]13[/latex].

The degree of the zero polynomial is defined to be [latex]-\infty[/latex].

Extra: Polynomials Over General Rings

This part is for the interested reader only. Most students can skip this part, or just remember that polynomials over [latex]\mathbb{C}[/latex] are the same as polynomials over [latex]\mathbb{R}[/latex], but with complex coefficients and that the degree of a monomial in more variables equals the sum of the exponents.

We have discussed polynomials over [latex]\mathbb{R}[/latex]. We shall later see that we can add, subtract and multiply these polynomials. In general, our coefficients [latex]c[/latex] do not need to belong to [latex]\mathbb{R}[/latex], but they can belong to any set of “numbers” in which we can add, subtract and multiply. These sets are called rings. Examples of rings are the real numbers [latex]\mathbb{R}[/latex], the integers [latex]\mathbb{Z}[/latex] and the complex numbers [latex]\mathbb{C}[/latex]. In this case, we talk about complex polynomials, or polynomials over [latex]\mathbb{C}[/latex]. The degree of a polynomial is defined in the same way as in the real case.

In particular, the polynomials over [latex]\mathbb{R}[/latex] form a ring, which we denote by [latex]\mathbb{R}[x][/latex]. The polynomials over this ring will be polynomials in two variables [latex]x[/latex] and [latex]y[/latex] over [latex]\mathbb{R}[/latex].

Here the degree in [latex]x[/latex] of [latex]x^3y^5[/latex]is [latex]3[/latex], the degree in [latex]y[/latex] of [latex]x^3y^5[/latex]is [latex]5[/latex] and its joint degree or degree is [latex]8[/latex].

Adding and Subtracting Polynomials

Polynomials can be added or subtracted by combining like terms.

Learning Objectives

Explain how to add and subtract polynomials and what it means to do so

Key Takeaways

Key Points

  • The rules for adding and subtracting algebraic expressions apply to polynomials; only like terms can be combined.
  • Any two polynomials can be added or subtracted, regardless of the number of terms in each, or the degrees of the polynomials.
  • The sum or difference of two polynomials will have the same degree as the polynomial with the higher degree in the problem.

Key Terms

  • Commutative Property: States that changing the order of numbers being added does not change the result.
  • degree of a polynomial: The highest value of an exponent placed on a variable in any of the terms of a polynomial.

Polynomials are algebraic expressions that contain terms that are constructed from variables and constants. Recall the rules for adding and subtracting algebraic expressions, which state that only like terms can be combined.

Like terms are those that are either both constants or have the same variables with the same exponents. For example, [latex]4x^3[/latex] and [latex]x^3[/latex]are like terms; [latex]21[/latex] and [latex]82[/latex] are also like terms. Adding and subtracting polynomials is as simple as adding and subtracting like terms. When adding polynomials, the commutative property allows us to rearrange the terms to group like terms together.

Note that any two polynomials can be added or subtracted, regardless of the number of terms in each, or the degrees of the polynomials. The resulting polynomial will have the same degree as the polynomial with the higher degree in the problem.

You may be asked to add or subtract polynomials that have terms of different degrees. For example, one polynomial may have the term [latex]x^2[/latex], while the other polynomial has no like term. If any term does not have a like term in the other polynomial, it does not need to be combined with any other term. It is simply carried down, with addition or subtraction applied appropriately. See the second example below for a demonstration of this concept.

Example 1

Find the sum of [latex]4x^2 - 5x + 1[/latex] and [latex]3x^2 - 8x - 9[/latex].

First, group like terms together:

[latex](4x^2 +3x^2 ) + (- 5x-8x) + (1 - 9)[/latex]

Combine the like terms for the solution:

[latex]7x^2 - 13x - 8[/latex]

Example 2

Subtract: [latex](5x^3 + x^2 + 9) - (4x^2 + 7x -3)[/latex]

Start by grouping like terms. Remember to apply subtraction to each term in the second polynomial. Note that the term [latex]5x^3[/latex] in the first polynomial does not have a like term; neither does [latex]7x[/latex] in the second polynomial. These are simply carried down.

[latex]5x^3 + (x^2 - 4x^2) + (- 7x) + (9 - (-3)) \\ 5x^3 + (x^2 - 4x^2) - 7x +(9 + 3)[/latex]

Now combine the like terms:

[latex]5x^3 - 3x^2 - 7x + 12[/latex]

Notice that the answer is a polynomial of degree 3; this is also the highest degree of a polynomial in the problem.

Multiplying Polynomials

To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.

Learning Objectives

Explain how to multiply polynomials using the distributive property and describe the results of doing so

Key Takeaways

Key Points

  • To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together.
  • To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.
  • The degree of a product of two polynomials equals the sum of the degrees of said polynomials.
  • The zeros of a product of two polynomial are the zeros of the two factors, combined.

Key Terms

  • monomial: An algebraic expression consisting of one term.
  • commutative: A binary operation is commutative if changing the order of the operands does not change the result, for example addition and multiplication.
  • polynomial: an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as [latex]a_n x^n + a_{n-1}x^{n-1} +… + a_0 x^0[/latex]. Importantly, because all exponents are positive, it is impossible to divide by [latex]x[/latex].

Multiplying a polynomial by a monomial is a direct application of the distributive and associative properties. Recall that the distributive property says that

[latex]a(b + c) = ab + ac[/latex]

for all real numbers [latex]a,b[/latex] and [latex]c.[/latex] The associative property says that

[latex](ab)c=a(bc)[/latex]

for all real numbers [latex]a,b[/latex] and [latex]c.[/latex]

As we will treat variables in the same way as real numbers, the same properties hold whenever [latex]a,b[/latex] and/or [latex]c[/latex] is a variable. So for the multiplication of a monomial with a polynomial we get the following procedure:

Multiply every term of the polynomial by the monomial and then add the resulting products together.

For example,

[latex]\begin{align} 3x^2(4x-\pi)&=3x^2 \cdot 4x-3x^2 \cdot \pi \\&=12x^3-3\pi x^2. \end{align}[/latex]

To multiply a polynomial [latex]P(x) = M_1(x) + M_2(x) + \ldots + M_n(x)[/latex] with a polynomial [latex]Q(x) = N_1(x) + N_2(x) + \ldots + N_k(x)[/latex], where both are written as a sum of monomials of distinct degrees, we get

[latex]\begin{align} P(x)Q(x) &= (M_1(x)+M_2(x)+ \ldots + M_n(x))Q(x) \\ &=M_1(x)Q(x) + M_2(x)Q(x) + \ldots + M_n(x)Q(x) \\ &=M_1(x)N_1(x) + \ldots + M_1(x)N_k(x) + \ldots \\ &+M_n(x)N_1(x) + \ldots + M_n(x)N_k(x) \end{align}[/latex]

and we see that this equals the sum of the products of the terms, where every term of [latex]P(x)[/latex] is multiplied exactly once with every term of [latex]Q(x)[/latex]. Notice that since the highest degree term of [latex]P(x)[/latex] is multiplied with the highest degree term of [latex]Q(x)[/latex] we have that the degree of the product equals the sum of the degrees, since

[latex]a^na^m=a^{m+n}[/latex]

for all real numbers (and variables) [latex]a[/latex] and all non-negative integers [latex]m[/latex] and [latex]n[/latex].

For convenience, we will use the commutative property of addition to write the expression so that we start with the terms containing [latex]M_1(x)[/latex] and end with the terms containing [latex]M_n(x)[/latex].

This method is commonly called the FOIL method, where we multiply the First, Outside, Inside, and Last pairs in the expression, and then add the products of like terms together.

For example, to find the product of [latex](2x+3)(x−4)[/latex], use FOIL and then add the products together:

[latex]\begin {align} 2x \cdot x+2x \cdot (−4)+3 \cdot x+3 \cdot(−4)&=2x^2-8x+3x-12 \\ &=2x^2−5x-12 \end {align}[/latex]

Zeros of a Product of Polynomials

Since we made sure that the product of polynomials abides the same laws as if the variables were real numbers, the evaluation of a product of two polynomials in a given point will be the same as the product of the evaluations of the polynomials:

[latex]P(x_0)Q(x_0) = PQ(x_0)[/latex]

for all real numbers [latex]x_0.[/latex]

In particular [latex]PQ(x_0) = 0[/latex] if and only if [latex]P(x_0)Q(x_0)=0[/latex], if and only if [latex]P(x_0) = 0[/latex] or [latex]Q(x_0) = 0[/latex]. So the roots of a product of polynomials are exactly the roots of its factors, i.e. [latex]x_0[/latex] is a zero for [latex]PQ(x)[/latex] if it is a zero for [latex]P(x)[/latex] or for [latex]Q(x)[/latex] (and possibly both).

Polynomial and Rational Functions as Models

Functions are commonly used in fitting data to a trend line. Polynomial and rational functions are both relatively accurate and easy to use.

Learning Objectives

Discuss the advantages and disadvantages of using polynomial and rational functions as models

Key Takeaways

Key Points

  • Researchers will often collect many discrete samples of data, relating two or more variables, without knowing the mathematical relationship between them. Curve fitting is used to create trend lines intended to fill in the points between and beyond collected data points.
  • Polynomial functions are easy to use for modeling, but ill-suited to modeling asymptotes and some functional forms, and they can become very inaccurate outside the bounds of the collected data.
  • Rational functions can take on a much greater range of shapes and are more accurate both inside and outside the limits of collected data than polynomial functions. However, rational functions are more difficult to use and can include undesirable asymptotes.

Key Terms

  • asymptote: A line that a curve approaches arbitrarily closely, as they go to infinity; the limit of the curve, its tangent “at infinity”.

For a set of data containing two (or more) quantitative properties, statisticians use functions to denote a relationship between two or more distinct properties. When we find a function that lies reasonably close to the collected data points, we create a trend line which says how one property behaves as a function of the other ones. We call this curve fitting. Then we can use this to make predictions about the former property once we know something about the others.

For example, if you measure the height of [latex]99[/latex] 10-year olds, and you measure how far they can jump and how fast they can run, you might be able to predict how fast the [latex]100th[/latex] kid can run, just by watching them jump. Of course, this is only possible if the two quantities are related: How many uncles a kid has got has probably nothing to do with how far they can jump.

Polynomial and rational functions are easy to work with because you only need to make use of elementary operations.

Recall that a polynomial function has the following form:

[latex]y=a_nx^n+a_{n-1}x^{n-1}+…+a_2x^2+a_1x+a_0[/latex]

where [latex]n[/latex] (the degree of the polynomial) is an integer greater than or equal to [latex]0[/latex], [latex]x[/latex] and [latex]y[/latex] are variables, and [latex]0\not=a_n,a_{n-1},\ldots,a_2, a_1[/latex] and [latex]a_0[/latex] are constants.

Polynomial functions are very simple in form and easy to use, but they have limitations with regard to statistical modeling. They can take on only a limited number of shapes and are particularly ill-suited to modeling asymptotes. They are finite when [latex]x[/latex] is finite and infinite when [latex]x[/latex] is infinite. It is especially hard to say something about the value of [latex]y[/latex] when [latex]x[/latex] is a lot bigger (smaller) than the greatest (smallest) [latex]x[/latex] in the dataset.

This need not be a problem. For lots of datasets, their are no asymptotes and data is more or less bounded. (For example, there are no humans who are [latex]-10[/latex] or [latex]10000[/latex] years old.)

Polynomials have also lots of advantages:

  • They are well studied: a lot of their properties are known.
  • They are easy to compute, using only multiplication and addition.
  • They are closed under rescaling or changing of locations: if we change a kilometer to a mile we still get a polynomial.

To deal with the asymptotic problems of polynomials, we also use rational functions:

A rational function is the ratio of two polynomial functions and has the following form:

[latex]y=\frac{a_nx^n+a_{n-1}+x^{n-1}+…+a_2x^2+a_1x+a_0}{b_mx^m+b_{m-1}+x^{m-1}+…+b_2x^2+b_1x+b_0}[/latex]

Here, n and m define the degrees of the numerator and denominator, respectively, and together, they define the degree of the polynomial. For example, if [latex]n=2[/latex] and [latex]m=1[/latex], the function is described as a quadratic/linear rational function.

Rational functions are a little more complex in form than polynomial functions, but they have an advantage in that they can take on a much greater range of shapes and can effectively model asymptotes. They are also more accurate than polynomial functions both inside and outside the limits of collected data. However, rational functions sometimes include undesirable asymptotes that can disrupt an otherwise smooth trend line. They are also a little bit more difficult to compute, since you also need division to compute them. However, they still run fast.

The sine function consists of several waves. As the degree of the polynomial increases, it is able to take a more and more similar shape to the sine wave. The fourth degree polynomial becomes fairly close compared to the others, having the correct number of peaks and troughs.

Curve fitting: Polynomial curves generated to fit points (black dots) of a sine function: The red line is a first degree polynomial; the green is a second degree; the orange is a third degree; and the blue is a fourth degree.

Polynomials and rational functions are used for approximation in many everyday devices. For example, every time we take a picture with a smartphone, our phone looks at some data points and fills in the appropriate colors in the blanks, thus saving us a lot of memory, with the help of rational functions. Every time we say something through the phone, our phone tries to reduce the background noise by approximating our sound for short periods of time, again with the help of rational functions.