Polynomials - Part I

By now we have seen how to solve equations with one or two unknown variables. In this chapter we will try and understand what are polynomials and how to perform mathematical operations (such as addition and subtraction) with the polynomials.

What we will study in this chapter :

1. Monomials
2. Binomials
3. Polynomials
4. Mathematical operations with polynomials.

1. Monomials 
Algebraic terms with single coefficient with variable (either single or multiple) are called monomials. For example, 3x, 9xyz,  y², xyz³ are all examples of monomials. But a division by a variable or inverse of a monomial  are not monomials. For example 1/x or z/ y²  are not monomials. (The reason for this will become more clear when we discuss binomials or polynomials later in the chapter). 
But   4  xyz  is a monomial.  
        5                                                                                                                               

Example 1 : multiply two monomials 10y * 2y. Verify your answer by putting y = 2

10y * 2 y = 20 y²

LHS : for y =2 is 20 * 4 = 80

RHS : for y = 2 is 20 * 4 = 80

Thus multiplication of two monomials 10 y and 2 y is 20 y² .

Example 2 : Can you multiply two monomials 2x², 3zy? Verify if your answer is correct by substituting x = 3, y = 5, z = 7

2x²  *  3zy  = 6x²zy

LHS : x = 3, y = 5, z = 7

= 2 (9) * 3 (7) * 5 =  1890

RHS = 6 * 9 * 7 * 5 = 1890

Thus multiplication of two monomials 2x² and 3zy is 6x²zy

Example 3 :  Can you add 10x + 3y?

No you cannot add algebraic expressions having different variables such as x and y as given. (10x + 3y) are called binomials, as we shall study later.

But if the problem was adding 10x and 3x then we could add the two monomials. The answer would have been 13x.

Example 4 :  Multiply three monomials 2x, 5x², 10x³. Verify your answer by taking the value of x to be 10.

2x  *  5x²  *  10x³  =  100x⁶                             (from the chapter on indices)

LHS : for x = 10

              = 2 * 10 * 5 * 100 * 10 * 1000  = 100000000

RHS = 100 * (10) ⁶    =  10⁸ = 100000000

Thus LHS = RHS. Hence the result of multiplying three monomials 2x, 5x², 10x³ is 100x⁶.  

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