Multiplying Polynomials

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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