Computing with Polynomials

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x3 − 5x + 8 is a polynomial equation, but x2 − 3/x + 5x3/2 is not a polynomial equation. In this article we shall discuss about computing polynomial addition, subtraction and multiplication problems. (Source: wikipedia)

Computing with polynomials example problem

Example:

Computing add of the polynomial 6x4 – 3x2 + 7x + 4 and 4x + 5x3 – 6x2 + 1.

Solution:

Use the associative properties and distributive properties add the polynomials

(6x4 – 3x2 + 7x + 4) + (5x3 – 6x2 + 4x +1)

= 6x4 + 5x3 – 3x2 – 6x2 + 7x + 4x + 4 + 1

= 6x4 + 5x3 – (3+6)x2 + (7+4)x + 5

= 6x4 + 5x3 – 9x2 + 11x + 5.

An another method is very helpful to adding two polynomials

6x4 +0x3– 3x2 + 7x + 4

0x4 + 5x3 – 6x2 + 4x +1

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6x4 + 5x3– 9x2 + 11x + 5

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Example:

Computing the product of polynomial x3 – 3x2 – 2 and 3x2 + 3x – 1.

Solution: (x3 – 3x2 – 2) (3x2 + 3x – 1)

= x3 (3x2 + 3x – 1) + (–3x2) (3x2 + 3x – 1) + (–2) (3x2 + 3x – 1)

= (3x5 + 3x4 – x3) + (–9x4 – 9x3 + 3x2) + (–6x2 – 6x + 2)

= 3x5 + 3x4 – x3 – 9x4 – 9x3 + 3x2 – 6x2 – 6x + 2

= 3x5 + (3x4 – 9x4) + (–9x3 – x3) + (3x2 – 6x2) + (–6x) +2

= 3x5 – 6x4 – 10x3 – 3x2 – 6x + 2.

Example:

Subtracting the polynomial 5x3 – 2x2 – 2 from 2x3 + 9x2 – 5x – 5.

Solution:

Using associative properties and distributive properties to subtract the given polynomials

(5x3 + 9x2 – 5x – 5) – (6x3 – 2x2 – 2)

= 5x3 + 9x2 – 5x – 5 – 6x3 + 2x2 + 2

= 5x3 – 6x3 + 9x2 + 2x2 – 5x – 5 + 2

= (5x3 – 6x3) + (9x2 + 2x2) + (–5x) + (–5+2)

= –x3 + 11x2 – 5x – 3

My forthcoming post is on Binary Number System and iseet 2013 syllabus will give you more understanding about Algebra.

Computing with polynomials practice problem

Problem:

Computing add of the polynomial 2x4 – 3x2 + 7x + 5 and 5x + 8x3 – 6x2 - 1.

Answer:

2x4 + 8x3– 9x2 + 12x + 4

Problem:

Computing the product of polynomial x3 – 2x2 – 2 and 4x2 + 2x – 2.

Answer:

4x5 – 6x4 – 6x3 – 4x2 – 4x + 4.