Polynomials
What is Polynomial
Polynomials are expressions which are composed of two algebraic terms. It is made up of two terms namely Poly (meaning “many”) and Nominal (meaning “terms.”).
Example: ax^{2} +bx + c (Quadratic polynomial).
Types of Polynomials
It is important to understand the terms before you learn about the polynomials types.
- Constants such as 1, 2, 3, etc.
- Variables such as g, h, x, y, etc.
- Exponents such as 5 in x^{5} etc.
There are three types of Polynomials based on the number of terms in it and those are:
- Monomial– having only one term. Example – 5x, 3, 6a^{4, }etc.
- Binomial– having two terms. Example – 5x+3, 6a^{4} + 17x
- Trinomial– having three terms. Example – 8a^{4}+2x+7, etc.
which can be combined using addition, subtraction, multiplication, and division but is never division by a variable.
Non Polynomial examples : 1/x+2, x^{-3}
Polynomial Function
A polynomial function is an expression constructed with one or more terms of variables with constant exponents. If there are real numbers denoted by a, then function with one variable and of degree n can be written as:
f(x) = a_{0}x^{n} + a_{1}x^{n-1 }+ a_{2}x^{n-2 }+ ….. + a_{n-2}x^{2 }+ a_{n-1}x + a_{n}
Polynomial Equations
The standard form of writing a polynomial function is to put the highest degree first then, at the last, the constant term. The standard form of the polynomial equation is given below:
y = a_{0}x^{n} + a_{1}x^{n-1 }+ a_{2}x^{n-2 }+ ….. + a_{n-2}x^{2 }+ a_{n-1}x + a_{n}
Example: b = a^{4} +3a^{3} -2a^{2} +a +1 is a polynomial equation
Solving Polynomials
It is easier to solve polynomials equation using the polynomial formula. Here is an example to show you can solve the two polynomials equation.
Polynomial Degree
A polynomial equation having one variable which has the largest exponent is called a degree of the polynomial.
Polynomial |
Degree |
Example |
Constant |
0 |
6 |
Linear |
1 |
3x+1 |
Quadratic |
2 |
4x^{2}+1x+1 |
Cubic |
3 |
6x^{3}+4x^{3}+3x+1 |
Quadratic |
4 |
6x^{4}+3x^{3}+3x^{2}+2x+1 |
Example – Find the degree of the polynomial 6s^{4}+ 3x^{2}+ 5x +19
Solution- The degree of the polynomial is 4.
Polynomial Operations
- Addition/Subtraction– the addition/subtraction of two or more polynomial always result in a polynomial
- Multiplication– Two or more polynomial when multiplied always result in a polynomial of higher degree (unless one of them is a constant polynomial).
- Division– Division of two polynomial may or may not result in a polynomial. Let us study below the division of polynomials in details.
Polynomial Division
If a polynomial has more than one term, we use long division method for the same. Following are the steps for it.
- Write the polynomial in descending order.
- Check the highest power and divide the terms by the same.
- Use the answer in step 2 as the division symbol.
- Now subtract it and carry down the next term.
- Repeat step 2 to 4 until you have no more terms to carry down.
- Note the final answer including remainder in the fraction form (last subtract term).
Polynomial Examples
Example- Given two polynomial 7s^{3}+2s^{2}+3s+9 and 5s^{2}+2s+1.
Solve these using mathematical operation. Solution- Given polynomial- 7s^{3}+2s^{2}+3s+9 and 5s^{2}+2s+1 Polynomial Addition– (7s^{3}+2s^{2}+3s+9) + (5s^{2}+2s+1) = 7s^{3}+(2s^{2}+5s^{2})+(3s+2s)+(9+1) = 7s^{3}+7s^{2}+5s+10 Hence addition result in a polynomial Polynomial Subtraction– (7s^{3}+2s^{2}+3s+9) – (5s^{2}+2s+1) = 7s^{3}+(2s^{2}-5s^{2})+(3s-2s)+(9-1) = 7s^{3}-3s^{2}+s+8 Hence addition result in a polynomial Polynomial Multiplication– (7s^{3}+2s^{2}+3s+9) × (5s^{2}+2s+1) = 7s^{3} (5s^{2}+2s+1)+2s^{2} (5s^{2}+2s+1)+3s (5s^{2}+2s+1)+9 (5s^{2}+2s+1)\) = (35s^{5}+14s^{4}+7s^{3})+ (10s^{4}+4s^{3}+2s^{2})+ (15s^{3}+6s^{2}+3s)+(45s^{2}+18s+9) = 35s^{5}+(14s^{4}+10s^{4})+(7s^{3}+4s^{3}+15s^{3})+ (2s^{2}+6s^{2}+45s^{2})+ (3s+18s)+9 = 35s^{5}+24s^{4}+26s^{3}+ 53s^{2}+ 21s +9 Polynomial Division– (7s^{3}+2s^{2}+3s+9) ÷ (5s^{2}+2s+1) (7s^{3}+2s^{2}+3s+9)/(5s^{2}+2s+1) This cannot be simplified. Therefore division of these polynomial do not result in a Polynomial. |