What is a Polynomial?
Before performing any operation on a polynomial, let us take a minute to first understand what a polynomial is? A polynomial can be defined as an algebraic expression whose power must be a non-integer. For example, a+bx+cx2+dx3+.......Here, a, b, c, and d are called constants and x is the variable.
Types of Polynomials
Classification on the basis of Terms
A polynomial can have one term, two-term, three or more than three terms:
Monomials – ‘Mono’ stands for one and ‘mial’ stands for terms thus an algebraic expression with one term is called a monomial. For example, 2x + 5x + 10x is a monomial because when we add the like terms it results in 17x. Furthermore, 4t, 21x2y, 9pq, etc are also monomials because each of these expressions has only one term.
Binomials – ‘Bi’ stands for two and ‘mial’ stands for terms therefore an algebraic expression with two, unlike terms are called binomials. For example, 3x + 4x2 and 10pq + 13p2q are binomials.
Trinomials – ‘Tri’ stands for three and ‘mial’ stands for terms thus an algebraic expression with three unlike terms are called trinomials. For example- 3x + 5x2 – 6x3 and 12pq + 4x2 – 10 are trinomials.
Classification on the Basis of Degrees
The Degree of Polynomial is the highest value of the exponent in the expression because it is the largest exponent. We can also call it an order of the polynomial. While we are finding the degree of the polynomial, we must remember that the polynomial powers of the variables should be either in ascending or descending order.
Linear Polynomial: If the expression is of degree 1 then it is called a linear polynomial. For Example 5x+2,50z+3.
Quadratic Polynomial: If the expression is of degree 2 then it is called a quadratic polynomial.For Example.
Cubic Polynomial: If the expression is of degree 3 then it is called a cubic polynomial.For Example.
Operations On Polynomials
We all are aware that there are four types of operations, that is, addition, subtraction, multiplication, and division. Even in case of a polynomial, we can do all the four operations. The process of operating one variable or term to another variable is simple and is called the operation of two monomials. But, the process of operating two or more variables remains the same as that of one variable, it's just that you need to keep a track of it.
There are Four Operations:
Addition - The process of finding the sum of the coefficient of two or more like terms is called the addition of polynomials.
Subtraction - The process of finding the sum of the coefficient of two or more like terms after changing the sign is called subtraction of polynomials.
Multiplication - The process of finding the product of terms is called multiplication of polynomials.
Division - The process of finding the quotient of polynomials is called the division of polynomials.
Multiplication of Polynomials
Multiplication of polynomials is a little challenging as compared to the process of adding and subtracting polynomials. According to distributive property, each term in the first polynomial is multiplied by each term in the second polynomial, and then the terms are combined. There is a short cut method called the FOIL method which is popular for the multiplication of binomials.
Types Of Multiplication
Multiplication Of Two Monomials
Multiplying polynomials is the process of multiplying monomials several times and keeping a track of it. In other words, the multiplication of polynomials is a combination of multiplication of several monomials.
For example; If x is a literal and m, n are positive integers, then xm x xn = xm+n
Therefore, the product of two monomials = (product of their numerical coefficient) x (product of their literal coefficients)
Multiplying Monomial by Binomial
When multiplying one-term by two terms, it is distributed into the parenthesis.
Example 1. Multiply 3x(4x+2y)
Solution
+3x will be multiplied with each term of the binomial that is +4x and +2y. Note: signs are also taken along with each term. Also, remember that two same signs result in a plus sign and two opposite signs result in a minus sign while finding the product of two terms.
So, the product of 3x(4x+2y) will be 12x2 + 6xy. Since there are no like terms to add together the answer will remain as it is.
Example 2. Multiply -3x(4x+2y).
Solution
-3x will be multiplied with each term of the binomial that is +4x and +2y. As mentioned above the signs are also taken along with each term. Two same signs result in a plus sign and two opposite signs result in a minus sign while finding the product of two terms. Since there is a negative sign with 3x the answer will be -12x2 -6xy.
Multiplication of Two Binomial - The FOIL Method
2 binomials can be multiplied using the Foil method. As discussed earlier, while multiplying the polynomial, each term of one polynomial needs to be multiplied by another term of the polynomial. The foil method is just a technique of remembering the series or order of finding the product.
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Method: Multiply the first terms then the outer terms then the inner terms and finally the last terms.
The product of two positives will be positive.
The product of two negatives will also be positive.
The product of a positive and negative will always be negative.
Example 3. (x+2) (x+1)
Solution
Multiply the first terms of both the binomials that are x from (x+2) and the x from (x+1). The product of x times x will be x2. Then we have to multiply the outer terms of both the binomials. The product of the outer terms that are the x from (x+2) and the 1 from (x+1) is x. Multiply the inner terms of the binomials. The inner terms here are the 2 from (x+2) and the x from (x+1) so the product is 2x. At last, multiply the last terms in each of the two binomials. The last terms here are the 2 from (x+2) and the 1 from (x+1) so the product is 2.
Final answer is x2 + x + 2x + 2
