What is a polynomial?
A polynomial is a mathematical expression with many terms ("poly" means "many" and "nomial" means "names" or "terms").
Polynomials can have any combination of operators (addition, subtraction, multiplication or division), constants, variables and integer exponents (powers), but never division by a variable. Remember, this means that expressions with negative indices can never be polynomials because $x^{-a}$x−a is the same as $\frac{1}{x^a}$1xa. Since the exponents for any variable must be integers, this also means that expressions with fractional exponents like $x^{\frac{1}{2}}$x12 are also not polynomials.
Polynomials are usually written in descending order, starting with the term with the highest power and ending with the term with the lowest (or no) power. For example, in the polynomial $8x+4x^8-2x^2+4$8x+4x8−2x2+4, the powers are all out of order. In descending order, this polynomial can be rewritten as $4x^8-2x^2+8x+4$4x8−2x2+8x+4.
| Examples of expressions that ARE polynomials | Examples of expressions that ARE NOT polynomials |
|---|---|
| $5x^2+\frac{4}{3}x-7$5x2+43x−7 | $\frac{4}{x-3}$4x−3 |
| $-18$−18 | $3+\frac{1}{x}$3+1x |
| $3x$3x | $4x^3-\frac{1}{x^7}+8$4x3−1x7+8 |
| $4c-8cd+2$4c−8cd+2 | $\frac{7}{8}x^{-2}+5$78x−2+5 |
| $22x^6+12y^8$22x6+12y8 | $\sqrt{x}$√x |
| $7g+\sqrt{12}$7g+√12 | $12f^3g^{-4}\times h^6$12f3g−4×h6 |
Parts of a polynomial
Variable: A variable is usually represented by a letter, such as $x$x , $y$y or $z$z, to represent an unknown quantity. For example, the variable $t$t in the polynomial $4t+3$4t+3 could represent time.
Multivariable: A polynomial or term with more than one variable. For example, $2xy+5$2xy+5 is a multivariable polynomial, while $3x^2+4x+1$3x2+4x+1 is not since it has only one variable ($x$x).
Term: A number, variable or a product of a number and variables where any exponents are positive integers. Examples of terms are $-2x$−2x, $\frac{5}{2}x^3$52x3 and $13$13. Note that $6x^{\frac{1}{2}}$6x12 is not a term because the exponent is not an integer.
Monomial: A polynomial with only one term. For example, $2x^2$2x2 is a monomial, as is $-4xy^3$−4xy3. A polynomial can also be binomial (two terms) or trinomial (three terms). If there are more than three terms, it is just referred to as a polynomial.
Constant term: The term in a polynomial that has no variables (that is, no algebraic terms). For example, in the polynomial $4y^8+2xy-4x-\frac{2}{3}$4y8+2xy−4x−23, the constant term is $-\frac{2}{3}$−23.
Coefficient: A constant which multiplies variables with positive integer powers. For example, in the monomial $-3xy^4$−3xy4, the variables are $x$x and $y$y while the coefficient is $-3$−3. In the monomial $x$x, the coefficient is $1$1.
Degree: The largest exponent (power) of a variable in a polynomial. For example, in the polynomial $x^3+4x^2-9$x3+4x2−9, the highest power of $x$x is $3$3, so the degree of this polynomial is $3$3. In a multivariable polynomial like $x^3y^8+2z^2$x3y8+2z2, the degree is the largest sum of exponents of the variables in any monomial (term). Here, the degree of $x^3y^8$x3y8 is $3+8=11$3+8=11, and the degree of $2z^2$2z2 is 2. So the degree of the polynomial $x^3y^8+2z^2$x3y8+2z2 is $11$11.
Leading coefficient: When a polynomial is written with its exponents in descending order, the leading coefficient is the number that is written before the first algebraic term. For example, in $5x-7$5x−7, the leading coefficient is $5$5. Sometimes you may need to use your knowledge of algebra to work out the leading coefficient. In the polynomial $-x^5-2x^4+4$−x5−2x4+4, the leading coefficient is $-1$−1. When the leading coefficient is 1, the polynomial is called a monic polynomial.
Combining like terms
Sometimes, you might need to combine “like terms” and simplify a polynomial before identifying the parts. Like terms are terms with the same variables and powers of these variables. The monomials $2xy$2xy and $5xy$5xy are like terms, while $5x^2y$5x2y and $2xy^2$2xy2 are not (the exponents of the variables are different).
Worked example
Example 1
Consider the polynomial $3-5x^2+4x+x^2+9-x$3−5x2+4x+x2+9−x . Identify the coefficient of $x^2$x2 and the constant in this polynomial.
Solution:
First, identify the like terms in the polynomial. Here, there are terms containing $x$x, $x^2$x2 and $x^0$x0 (constant terms). Group them together and simplify:
| $3-5x^2+4x+x^2+9-x$3−5x2+4x+x2+9−x | $=$= | $-5x^2+x^2+4x-x+3+9$−5x2+x2+4x−x+3+9 |
| $=$= | $-4x^2+3x+12$−4x2+3x+12 |
The coefficient of $x^2$x2 is $-4$−4. The constant term is $12$12.
Polynomials are any combination of operators (addition, subtraction, multiplication or division), constants, variables and integer exponents.
The degree of a single variable polynomial is the largest exponent of that variable. For a multivariable polynomial, the degree is the largest sum of exponents of the variables in any term.
Practice questions
Question 1
Is $2x^3-4x^5+3$2x3−4x5+3 a polynomial?
Yes, it is a polynomial.
ANo, it is not a polynomial.
B
Question 2
For the polynomial $P\left(x\right)=\frac{x^7}{5}+\frac{x^6}{6}+5$P(x)=x75+x66+5
The degree of the polynomial is: $\editable{}$
The leading coefficient of the polynomial is: $\editable{}$
The constant term of the polynomial is: $\editable{}$
Question 3
Consider the polynomial $12+7p^2$12+7p2.
What is the coefficient of $p$p in this polynomial?
$7$7
A$2$2
B$12$12
C$0$0
D