An Overview of Polynomials
In algebra, you may have studied variables such as x and y. Many expressions involve variables and constants. They are called polynomials. Polynomials are used in various fields such as science, technology, economics, social sciences, etc. These can be used to model a problem that occurs in nature, business, etc., as equations. These models can be used to find solutions to complex scientific problems. We can also classify polynomials based on the number of terms. We will discuss about different types of polynomials in this article.
What is Monomial?
A monomial is a type of polynomial that has only a single term. It can be a variable with a constant coefficient or only a constant. The term should not be zero. If a polynomial has only a constant, it is called a constant polynomial. A monomial should also have a non-negative degree, where the degree is the sum of exponents of variables in a term.
Examples of monomials are as follows:
\[5\]
\[3x\]
\[4y\]
\[16xy\]
\[2mn\]
What is Binomial?
A binomial is a polynomial that has only two non-zero terms related by addition or subtraction. A constant term can be present in a binomial. Examples of binomials are as follows:
\[x\, + \,3\]
\[3x\, + \,4\]
\[4xy\, + 3y\]
What is Trinomial?
A trinomial is a polynomial that has three terms related by addition or subtraction. All of its terms should not be zero. There can be a constant term in a trinomial.
Examples of trinomials are as follows:
\[3{x^2} + 4x + 9\]
\[6{x^2} + 3xy + 9{y^2}\]
What is Polynomial?
A polynomial is an expression consisting of variables and constants, related by addition or subtraction. It should have more than a single term. Examples of polynomials are as follows:
\[2{x^2} + 15x + 7\]
\[12{x^2} + 4x + y\]
Terms of a Polynomial
Terms are defined as variables with co-efficient or constants related by addition or subtraction. Polynomials are classified as monomials, binomials and trinomials based on the number of terms. Two or more variables can also be multiplied together in a single term.
In the following example, the terms are:
Polynomial: \[3 + 10xy + 9x{y^2} + 16xyz\]
Terms: \[3,\,\,10xy,\,\,9x{y^2},\,\,16xyz\]
Factors of Terms of a Polynomial
Factors are constants or variables that can divide a term without leaving any remainder. For example, for the term \[5xy\], factors are \[5,\,x,\,y.\] Factorization of polynomials is done to solve polynomial equations.
Degree of a Polynomial
It is defined as the highest power of variables in the terms of a polynomial. The powers of all variables in a term should be added together to calculate the degree of the term. Then, the degrees of all terms of a polynomial should be compared to find the degree of the polynomial. If the degree of a polynomial is \[2\], it is called a quadratic polynomial. It has \[2\]roots. If the degree of a polynomial is \[3\], it is called a cubic polynomial, and it has \[3\] roots. For example, the degree of \[3{x^2} + 2\] is \[2\].
Addition and Subtraction of Polynomials
We can add or subtract like terms in two or more polynomials. Like terms are terms with the same variables in them. For example, if we add \[3x + 4\] and \[4x + 7\], we get \[(3x + 4) + (4x + 7)\,\, = \,\,(3x\, + \,4x)\, + \,(4\, + 7)\,\, = \,\,7x + 11\].
For the subtraction of \[3x + 4\] from \[4x + 7\],
\[(4x + 7)\,\, - \,\,(3x + 4)\,\, = \,\,(4x\, - \,3x)\,\, + \,\,(7\, - \,4)\, = \,x\, + \,3\]
Multiplication of a Polynomial with a Constant
To multiply a polynomial by a constant, we should multiply each term of the polynomial by the constant. For example, if we like to multiply \[3x + 5\] by 4, we should multiply each term by \[4\], \[4\,\,\times \,\,(3x\, + \,5) = \,12x\, + \,20\].
Interesting Facts
In algebra, polynomials are used to find unknown values in calculations.
Polynomials are used to study the relationship between two or more variables.
They can also be used to find the maximum and minimum value of a variable in a specified interval.
Solved Problems
Classify the following polynomials as monomial, binomial and trinomial.
\[\begin{array}{l}3x & & & 5x + 3\\45 & & & 3x + 4y\\7{x^3} & & & 2xy + 3{x^2} + {y^3}\\6{x^2} & & & 6{x^3} + 3x{y^2} + 4{y^4}\end{array}\]
Ans:
Monomials: \[3x,\,45,\,\,7{x^3},\,\,6{x^2}\]
Binomials: \[5x + 3,\,\,3x + 4y\]
Trinomials: \[2xy + 3{x^2} + {y^3},\,\,\,6{x^3} + 3x{y^2} + 4{y^4}\]
Find the degree of the polynomials below.
\[4{x^3} + 5x + 6{x^2}\]
Ans:
Since \[3\] is the highest among the degrees of all the terms, the degree of the polynomial is \[3\].
b) \[2x{y^2} + 15xy + 30x\]
Let us consider the term \[2x{y^2}\].
Here, the exponent of x = \[1\]
Exponent of y = \[2\]
The degree of the term, \[2x{y^2}\] = Exponent of x + Exponent of y
= \[1\, + \,2\]
= \[3\]
Similarly, we can find the degrees of other terms.
Since \[3\] is the highest among the degrees of all the terms, the degree of the polynomial is \[3\].
3. Add the polynomials \[4{x^2} + 3x + 4\] and \[5x + 3\].
Ans:
Adding the like terms,
Addition of Polynomials
Practice Questions
Write the degree of the following polynomials.
\[4{x^3} + 2{x^2} + 5,\,\,2{x^2} + 3x,\,\,3xy,\,\,5x{y^2}\]
(Ans: 3, 2, 2, 3)
Add the polynomials \[3x + 2\] and \[5x + 4\].
(Ans: \[8x + 6\])
Conclusion
In this article, we learned about monomials, binomials, trinomials, and polynomials, with examples, and factors of polynomials. Polynomials are an important part of Mathematics that helps us deal with unknown quantities easily using mathematical expressions. We have also read about the applications of polynomials.
FAQs on Monomials, Binomials, Trinomials and Polynomials
1. What are monomials, binomials, and trinomials? Provide an example for each.
Monomials, binomials, and trinomials are types of polynomials classified by the number of terms they contain:
- A monomial is an algebraic expression with only one term. Example: 7x, 5y², or 12.
- A binomial is an algebraic expression with two unlike terms. Example: 3a + 4b, or x² - 9.
- A trinomial is an algebraic expression with three unlike terms. Example: x² + 2x - 1.
Any expression with one or more terms is broadly called a polynomial.
2. How can you identify if an algebraic expression is a polynomial?
To be considered a polynomial, an algebraic expression must follow two key rules:
- The exponents of the variables must be non-negative integers (0, 1, 2, 3,...). For example, x⁻² or √x (which is x¹/²) are not allowed.
- The variables cannot be in the denominator. For instance, an expression like 5/y is not a polynomial because it is equivalent to 5y⁻¹, which has a negative exponent.
3. What is the difference between a 'term' and a 'coefficient' in a polynomial?
In a polynomial, terms are the individual parts of the expression that are separated by addition (+) or subtraction (-) signs. The coefficient is the numerical part of a term that is multiplied by the variable. For example, in the polynomial 8x² - 3y + 5:
- The terms are 8x², -3y, and 5.
- The coefficient of the x² term is 8.
- The coefficient of the y term is -3.
4. What does the 'degree' of a polynomial signify?
The degree of a polynomial is the highest exponent of the variable in any of its terms. It helps classify the polynomial. For example:
- In the monomial 6x⁵, the degree is 5.
- In the trinomial 4x³ + 2x² - 9x, the highest power is 3, so the degree of the polynomial is 3.
5. How are 'like terms' and 'unlike terms' important when adding or subtracting polynomials?
Like terms are terms that have the same variables raised to the same power (e.g., 3x² and -5x²). Unlike terms have different variables or different powers (e.g., 3x² and 3x). This distinction is crucial because you can only combine (add or subtract) like terms. For example, to simplify (2x² + 5x) + (4x² - 2x), you combine the like terms: (2x² + 4x²) + (5x - 2x) = 6x² + 3x.
6. Why is an expression with a variable in the denominator, like 7/x, not considered a polynomial?
An expression like 7/x is not a polynomial because it violates the fundamental rule for exponents. According to the laws of exponents, 7/x can be written as 7x⁻¹. A key condition for an expression to be a polynomial is that all variables must have non-negative integer exponents. Since the exponent here is -1, the expression is not a polynomial.
7. Can a polynomial have more than three terms? If so, what is it called?
Yes, a polynomial can have any number of terms. While we have special names for expressions with one (monomial), two (binomial), or three (trinomial) terms, an expression with four or more terms is generally just called a polynomial. The term 'polynomial' itself means 'many terms' and is the umbrella category for all these expressions.
8. Is a single number, like 15, considered a monomial or a polynomial?
Yes, a single number like 15 is considered both a monomial and a polynomial. It is called a constant polynomial. We can think of it as 15x⁰, since any variable raised to the power of zero is 1. Because it has only one term (15), it qualifies as a monomial. The degree of a constant polynomial is 0.
9. How does the degree of a polynomial help in its classification?
The degree of a polynomial is used for classification beyond just the number of terms. It describes the nature of the polynomial, which is especially important when graphing. The most common classifications by degree are:
- Degree 1: Linear Polynomial (e.g., 3x + 2)
- Degree 2: Quadratic Polynomial (e.g., 5x² - x + 4)
- Degree 3: Cubic Polynomial (e.g., 2x³ + 7x)
10. What does it mean for a value to be a 'zero' of a polynomial?
A 'zero' of a polynomial is any value of the variable that makes the entire polynomial equal to zero. It is also known as a 'root' of the polynomial equation. For example, in the binomial P(x) = x - 4, the value x = 4 is a zero because when you substitute it, P(4) = 4 - 4 = 0. Finding the zeros of a polynomial is a fundamental concept in algebra.
