The product of a polynomial and a scalar is obtained by multiplying each term of the polynomial by the scalar.

Example;

If p(x) and g(x) are polynomials where,

P(x) = 2x³ - 3x² + 5x – 10,  g(x) = 3x² – 6x + 5

Find;

1. 2p(x)   (ii) -3g(x)

Solution

1. 2p(x) = 2 [ 2x³ – 3x² + 5x -10]

           = 4x³-6x²+10x-20

2. -3g(x) = -3 [ 3x² – 6x + 5]

           = -9x²+18x-15

Multiplication of a polynomial by another polynomial

Recall:

(a + b) (2a + 3b) = 2a² + 3ab + 2ab + 3b²

                             = 2a² + 5ab + 3b²

As seen above, multiplication of two polynomials are done similar. Multiplying all terms and adding similar power terms together.

Example; given that

P(x) = 2x³ + 3x² – 2x + 1   , h(x) = (2x + 3), find; h(x) × p(x)

Solution

h(x) × p(x) = (2x + 3) (2x³ + 3x² – 2x + 1)

                   = (4x⁴ + 6x³ – 4x² + 2x) + (6x³ + 9x² – 6x + 3)

                  = 4x⁴ + 12x³ + 5x² – 4x + 3