What we discuss here:
- Meaning of the terms of an expression
- What is a polynomial and what is not a polynomial
- Degree of a polynomial
- Like and unlike terms
- How to add, subtract, and multiply polynomials
Terms of an Expression
The parts of an expression connected by the signs + or – are called the terms of the expression. For example, $x^{2}$, $2xy$, and $\sqrt{x}$ are the terms of the expression $$x^{2}+2xy+\sqrt{x},$$ and the expression $$2a-ab+c^{2}$$ is made up of three terms: $2a$,$-ab$, and $c^{2}$.
Polynomials
A polynomial or more precisely a polynomial in $x$ is an algebraic expression consisting of terms in the form $$ax^{k}$$ where $k$ is a nonnegative integers (that is zero or a natural number 0, 1, 2, …) and $a$ is a real number called the coefficient of the term. For example,
\[3,\quad x,\quad7x^{51},\quad-x^{3}+\frac{1}{\sqrt{14}}x-3,\quad\text{ and}\quad 7x^{5}-\pi x^{4}-\sqrt{2}x^{3}-x^{2}+x-3\]
are all polynomials. In the last example, the coefficients are $7,-\pi,-\sqrt{2},-1, 1,$ and $-3$.
- Note that any constant is also a polynomial because it can be written as $ax^{0}$; for example $3=3x^{0}$.
- Of course, instead of a polynomial in $x$, we may have polynomials in $y$ or $z$ or any other letter. For example, $4z^{3}-0.5z+1$ is a polynomial in $z$.
Monomials, Binomials, and Trinomials
- A polynomial that has only one term is called a monomial. For example, $4x^{3}$ and $-\sqrt{7}x^{2}$ are two monomials.
- A polynomial that has two terms is called a binomial. For example, $3x+2$ and $-4x^3+x$ are two binomials.
- A polynomial that has three terms is called a trinomial. For example, $3x+8x^5-9$ is a trinomial.
Degree of a Polynomial
The largest exponent in a polynomial is called the degree of the polynomial. In $$7x^{5}-\pi x^{4}-\sqrt{2}x^{3}-x^{2}+x-3,$$ the largest power of $x$ and hence the degree of the polynomial is 5.
| name | form | degree |
| Constant polynomial | $a$ | 0 |
| Linear polynomial | $ax+b$ | 1 |
| Quadratic polynomial | $ax^2+bx+c$ | 2 |
| Cubic polynomial | $ax^3+bx^2+cx+d$ | 3 |
Basic Operations on Polynomial
- When two or more terms differ only in their numerical coefficients, we say they are similar or like terms. For example, $4x^{2}$, $\frac{3}{2}x^{2}$, and $-\sqrt{3}x^{2}$ are like terms but $3x$ and $3x^{2}$ are unlike terms.
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Adding and subtracting polynomials: To add or subtract polynomials, we add or subtract the coefficients of like terms. For example:
\begin{align*} (7x^{3}-6x^{2}+4x+9)+(x^{3}+4x^{2}-2) & =(7+1)x^{3}+(-6+4)x^{2}+(4+0)x+(9-2)\\ & =8x^{3}-2x^{2}+4x+7 \end{align*}
and
\begin{align*} (-3x^{4}-8x)-(5x^{3}-3x+\sqrt{2}) & =(-3-0)x^{4}+(0-5)x^{3}+(-8-(-3))x+(0-\sqrt{2})\\ & =-3x^{4}-5x^{3}-5x-\sqrt{2} \end{align*}
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Multiplication of polynomials: We multiply polynomials like any other sums and simplify the result by using the exponent rule $$ax^{n}\cdot bx^{m}=ab\ x^{m+n}$$ and collect the like terms. For example,
\begin{align*} (4x^{2}-3x+5)(2x^{3}-x) & =4x^{2}(2x^{3}-x)-3x(2x^{3}-x)+5(2x^{3}-x)\\ & =8x^{5}-4x^{3}-6x^{4}+3x^{2}+10x^{3}-5x\\ & =8x^{5}-6x^{4}+(10-4)x^{3}+3x^{2}-5x\\ & =8x^{5}-6x^{4}+6x^{3}+3x^{2}-5x \end{align*}
It is clear that if $A$ is a polynomial of degree $n$ and $B$ is a polynomial of degree $m$, then the product $A\cdot B$ is a polynomial of degree $n+m$. In the above example, the first one is a polynomial of degree 2 and the second one is a polynomial of degree 3, and the product is a polynomial of $5=2+3$.
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Division of polynomials: To divide one polynomial by another, we use long division; the procedure for such a division has been explained in detail in the next Section