Standard form and types of polynomials

In the last les­son we sorted expres­sions by the num­ber of terms they have. Math­e­mati­cians more often sort poly­no­mi­als by their degree, because the degree decides how a poly­no­mial behaves: how many zeros it can have, how it divides, and how it fac­torises. This les­son sets out the stan­dard way of writ­ing a poly­no­mial in one vari­able and the names given to poly­no­mi­als of each degree, includ­ing two spe­cial cases, the con­stant poly­no­mial and the zero poly­no­mial.

Stan­dard form of a poly­no­mial#

An alge­braic expres­sion of the form

$$a_nx^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \dots + a_1x + a_0$$

where $a_n \neq 0$ and $n$ is a whole num­ber ($n \in W$), is called a poly­no­mial in $x$ of degree $n$. We usu­ally name it $p(x)$, $f(x)$ or $g(x)$.

The num­bers $a_n, a_{n-1}, a_{n-2}, \dots, a_1, a_0$ are the coef­fi­cients of the terms. Two of them have spe­cial names:

• $a_nx^n$, the term with the high­est power, is the lead­ing term, and $a_n$ is the lead­ing coef­fi­cient. It must not be zero, oth­er­wise the degree would not be $n$.
• $a_0$ is the con­stant term.

Writ­ing a poly­no­mial in stan­dard form means arrang­ing its terms in descend­ing pow­ers of $x$.

Exam­ple 1#

Write $5 - 3x^2 + 7x^4 - x$ in stan­dard form and state its degree, lead­ing coef­fi­cient and con­stant term.

Arrange the pow­ers from high­est to low­est:

$$7x^4 - 3x^2 - x + 5$$

There is no $x^3$ term, which is the same as a coef­fi­cient of $0$ for $x^3$. The degree is $4$, the lead­ing coef­fi­cient is $7$ and the con­stant term is $5$.

Poly­no­mi­als of var­i­ous degrees#

The con­di­tion $a \neq 0$ appears every time for the same rea­son: if the lead­ing coef­fi­cient were $0$, that term would van­ish and the degree would drop.

Lin­ear poly­no­mi­als can have more than one vari­able. $x + y$ and $y + z - 1$ are lin­ear poly­no­mi­als, because every term has degree at most $1$. The gen­eral form $ax + b$ in the table is for one vari­able.

Notice also that a qua­dratic does not need all three terms. $x^2 + 5$ has no $x$ term and $2x^2$ has nei­ther an $x$ term nor a con­stant, but each has degree $2$, so each is qua­dratic.

Con­stant poly­no­mial#

A poly­no­mial of degree $0$ is a con­stant poly­no­mial. Exam­ples are $5$, $2\sqrt{2}$ and $\displaystyle -\frac{1}{2}$. The degree is $0$ because

$$5 = 5 \times 1 = 5 \times x^0$$

Zero poly­no­mial#

If all the coef­fi­cients of a poly­no­mial are zero, it is called the zero poly­no­mial. For instance, $p(x) = ax^2 + bx + c$ is the zero poly­no­mial when $a = b = c = 0$, and then $p(x) = 0$.

The degree of the zero poly­no­mial is not defined. The rea­son is that $0$ can be writ­ten with any power of $x$ we like:

$$0 = 0 \times x^0, \qquad 0 = 0 \times x^1, \qquad 0 = 0 \times x^2, \qquad \dots$$

so there is no sin­gle power we could call its degree.

Remem­ber. The con­stant poly­no­mial $5$ has degree $0$. The zero poly­no­mial $0$ has no degree at all. These are dif­fer­ent state­ments.

Multi­n­o­mi­als#

An alge­braic expres­sion con­tain­ing one or more terms is called a multi­n­o­mial. Exam­ples are

$$\displaystyle 2 + \frac{1}{x}, \qquad 3 + 4x + 7x^2, \qquad 3 + \sqrt{x}$$

Every poly­no­mial is a multi­n­o­mial, but a multi­n­o­mial need not be a poly­no­mial. In the list above only $3 + 4x + 7x^2$ is a poly­no­mial; the other two have neg­a­tive or frac­tional pow­ers.

Exam­ple 2#

Which of the fol­low­ing is a poly­no­mial? (1) $x^2 - 6x + y$ (2) $5x^{-6} - 9x^2 - 7$ (3) $\displaystyle 5x^{\frac{1}{3}} - 9$ (4) $6\sqrt{x} - 10$

Only (1). The oth­ers con­tain the pow­ers $-6$, $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{1}{2}$, none of which is a whole num­ber.

Exam­ple 3#

If $ax^{n-1} + bx^{n-2} + cx^{n-3}$ is a cubic poly­no­mial, where $n \in N$, find $n$.

The high­est power is $n - 1$, and a cubic has degree $3$, so

$$n - 1 = 3 \implies n = 4$$

Exam­ple 4#

If $(a - 2)x^2 + (b - a)x + (c + b)$ is the zero poly­no­mial, find $a$, $b$ and $c$.

Every coef­fi­cient must be zero.

$$a - 2 = 0 \implies a = 2$$

$$b - a = 0 \implies b - 2 = 0 \implies b = 2$$

$$c + b = 0 \implies c + 2 = 0 \implies c = -2$$

So $a = 2$, $b = 2$ and $c = -2$. Each answer feeds the next equa­tion, so solve them in order.

Prac­tice#

1. Write $4 - 2x + 6x^3$ in stan­dard form and state its degree.
2. Clas­sify each poly­no­mial as lin­ear, qua­dratic, cubic, bi-qua­dratic or con­stant: (a) $x + x^2 + 4$ (b) $3x - 2$ (c) $7t^4 + 4t^3 + 3t - 2$ (d) $-5$ (e) $2x^3 - 7x + 4$ (f) $t^2 + 1$
3. Find the lead­ing coef­fi­cient and the con­stant term of $3 - 5x^2 + 8x^3 - x$.
4. Write the gen­eral form of a cubic poly­no­mial in $y$.
5. If $3x^{n+1} + 5x^n - 2$ is a bi-qua­dratic poly­no­mial, find $n$.
6. If $(k - 2)x^3 + 5x^2 - x + 1$ is a qua­dratic poly­no­mial, find $k$.
7. If $(p - 3)x^2 + (q + 1)x + (r - q)$ is the zero poly­no­mial, find $p$, $q$ and $r$.
8. Which of these is a multi­n­o­mial but not a poly­no­mial? (a) $x^2 + 1$ (b) $\displaystyle 2 + \frac{1}{x}$ (c) $7$
9. What is the degree of (a) the con­stant poly­no­mial $7$ and (b) the zero poly­no­mial?

Answers#