What is a polynomial?
A polynomial is a mathematical expression with many terms ("poly" means "many" and "nomial" means "names" or "terms").
A polynomial can have any combination of operators (addition, subtraction, multiplication or division), constants, variables and integer exponents, 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.
Mathematicians like order so 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 jumbled. To put it in order, we would rewrite it 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
Degree: The largest exponent (i.e. power) of a variable in a polynomial. e.g. 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.
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. e.g. In $-x^5-2x^4+4$−x5−2x4+4, the leading coefficient is $-1$−1. When the leading coefficient is 1, the polynomial is called a monic polynomial.
Constant term: the term in a polynomial that has no variables (i.e. no algebraic terms). e.g. in the polynomial $4y^8+2xy-4x-\frac{2}{3}$4y8+2xy−4x−23, the constant term is $-\frac{2}{3}$−23.
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