## Multiplying Polynomials

When we multiply a polynomial by a constant term, then each term of the polynomial is multiplied by that constant. And, in case of multiplying the two polynomials, each term of one polynomial is multiplied by every term of the another polynomial, and simplify the result.

Let f(x) and g(x) be two polynomials in the same variable x defined by
f(x)= a0 + a1x + a2x2 + …………… + anxn
g(x)= b0 + b1x + b2x2 + …………… + bnxn

Then the product of two polynomials is
f(x) × g(x) = a0b0 + (a0b1 + a1b0)x + (a0b2 + a2b0 + a1b1)x2 + ……………
Which has the degree n + n = 2n

Therefore the degree of the product of two polynomials is equal to the sum of the degrees of the two polynomials.

To find the product of two polynomials, we multiply each term of the polynomial by every term of the second and then combine like terms and simplify and arrange the terms in ascending or descending order.

Example:
Let f(x) = 4x2 + 2x and g(x) = x2 + x + 5
Then, f(x) × g(x) = (4x2 + 2x)(x2 + x + 5)
= 4x2(x2 + x + 5) + 2x(x2 + x + 5)
= 4x4 + 4x3 + 20x2 + 2x3 + 2x2 + 10x
= 4x4 + 6x3 + 22x2 + 10x

### Cancellation Law

Let f(x), g(x) and h(x) be three polynomials such that f(x) × g(x) = h(x) × g(x). Then, f(x) = h(x). This is called cancellation law.

### Properties of Multiplication of polynomials

Multiplication of the polynomials follows the following properties:
1) Closure property: If f(x) and g(x) be two polynomials then their product f(x).g(x) is also a polynomial.
2) Commutative property: If f(x) and g(x) be two polynomials, then f(x).g(x) = g(x).f(x).
3) Associative property: If f(x), g(x) and h(x) be three polynomials, then f(x).[g(x).h(x)] = [f(x).g(x)].h(x).
4) Multiplicative identity: For any polynomial f(x) there is the unit polynomial i(x) = 1x0 = 1 such that f(x).1 = f(x). Here, 1 is said to be the multiplicative identity.

### Workout Examples

Example 1: Find the product of the given polynomials.
a.     3x3 + 5x2 + 9x + 8    and     3x + 4
b.    4x2 + 5x + 7               and      x + 1
c.     3x3 – 2x2 + 5x – 1     and      2x – 3

Solution:  a) Let, f(x) = 3x3 + 5x2 + 9x + 8     and     g(x) = 3x + 4

f(x) × g(x) =  (3x3 + 5x2 + 9x + 8)(3x + 4)
= 3x3(3x + 4) + 5x2(3x + 4) + 9x(3x + 4) + 8(3x + 4)
= 9x4 + 12x3 + 15x3 + 20x2 + 27x2 + 36x + 24x + 32
= 9x4 + 27x3 + 47x2 + 60x + 32

b) Let, f(x) = 4x2 + 5x + 7     and     g(x) = x + 1

f(x) × g(x) =  (4x2 + 5x + 7)(x + 1)
= 4x2(x + 1) + 5x(x + 1) + 7(x + 1)
= 4x3 + 4x2 + 5x2 + 5x + 7x + 7
= 4x3 + 9x2 + 12x + 7

c) Let, f(x) = 3x3 – 2x2 + 5x – 1     and     g(x) = 2x – 3

f(x) × g(x) =  (3x3 – 2x2 + 5x – 1)(2x – 3)
= 3x3(2x – 3) – 2x2(2x – 3) + 5x(2x – 3) -1(2x – 3)
= 6x4 – 9x3 – 4x3 + 6x2 + 10x2 – 15x – 2x + 3
= 6x4 – 13x3 + 16x2 – 17x + 3

You can comment your questions or problems regarding the multiplication of polynomials here.