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Multiplying Polynomials
  • Difficulty Level : Basic
  • Last Updated : 19 Jan, 2021

Monomial is an algebraic expression that contains only one term. Monomial can be a combination of numbers and variables. Example of monomial expression is 5, 1526x, 1526xyz, 2x2, etc. Whereas Polynomial is made of two terms Poly meaning “Many” and Nomial which means “terms”. Therefore polynomial means a combination of many terms is called a polynomial. Polynomial is a combination of constants, variables, and exponents which are related using mathematics operations such as addition, subtraction, multiplication, etc. Example of polynomial are x2 + 5x + 26, x4 + 5x3 + 2x2 + 6x + 1 etc.

Monomial Multiplied by a Monomial

A monomial multiplied by a monomial or constant is also a monomial.

Examples:

i) 5 * 5 = 25 (constant multiply by constant)

ii) 5 * x = 5x (constant multiply by monomial)



iii) 5x * y = 2xy (monomial multiply by monomial)

iv) 2x * 2z = 4xz ( monomial multiply by monomial)

v) 6xz * y = 6xyz ( monomial multiply by monomial)

Monomial Multiplied by a Polynomial

To multiply a polynomial and a monomial we need to multiply each and every term of the polynomial with monomial.

Examples:

i) 5x * (5x2 + 2x + 6) =  (5x * 5x2) + (5x * 2x) + (5x * 6)

                                 = 25x3 + 10x2 + 30x

ii) 5 * (x4 + 2x + 6) = (5 * x4)+ (5 * 2x) + (5 * 6)

                               = 5x4 + 10x + 30

iii) z * (5xy + 2y + 6) =  (z * 5xy) + (z * 2y) + (z * 6)

                                 = 5xyz + 2yz + 6z

iv) xy * (4z + 1) = (xy * 4z) + (xy * 1)

                         = 4xyz + xy

Polynomial Multiplied by a Polynomial

To multiply a polynomial and a monomial we need to multiply each and every term of one polynomial with each and every term of other polynomials.

Examples:

i) (5x2 + 2x + 6) * (1x2 + 2x + 3) 

   = (5x2 * 1x2) + (5x2 * 2x) + (5x2 * 3) + (2x * 1x2) + (2x * 2x) + (2x * 3) + (6 * 1x2) + (6 * 2x) + (6 * 3)

   = 5x4 +10x3 + 15x2 + 2x3 + 4x2 + 6x + 6x2 + 12x + 18

   = 5x4 +12x3 + 21x2 + 18x + 18

ii) (3x2 + 1x + 2) * (1x2 + 2x + 1) 

    = (3x2 * 1x2) + (3x2 * 2x) + (3x2 * 1) + (1x * 1x2) + (1x * 2x) + (1x * 1) + (2 * 1x2) + (2 * 2x) + (2 * 1)

    = 3x4 +6x3 + 3x2 + 1x3 + 2x2 + 1x + 2x2 + 4x + 2

    = 3x4 +7x3 + 7x2 + 5x + 2

iii) (5xy + 1) * (2z + 3) = (5xy * 2z) + (5xy * 3) + (1 * 2z) + (1 * 3)

                                     = 10xyz +15xy + 2z + 3

iv) (3xyz) * (2x + 6) = (3xyz * 2x) + (3xyz * 6)

                                = 6x2yz +18xyz

Example of Multiplication of Algebraic Expressions

i) (−a3b) * (2ab3) = -2a4b4

ii) ((4 * 3) * (x * x2)) * (y + 2) = ((12) (x3)) * (y + 2)

                                             = (12x3) * (y + 2)

                                             = (12x3y + 30x3)

iii) (x2 + 2x + 4) * (x + 1) = (x2 * x) + (x2 * 1) + (2x * x) + (2x * 1) + (4 * x) + (4 * 1)

                                         = x3 + x2 + 2x2 + 2x + 4x + 4

                                         = x3 + 3x2 + 6x + 4

iv) (xy + 2y) * (a + b) = (xy * a) + (xy * b) + (2y * a) + (2y * b)

                                  = axy + bxy + 2ay + 2by

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