Solving Polynomial Equations: A Detailed Guide with Examples

Polynomial equations are a fundamental part of algebra. This article focuses on a specific type of polynomial equation, and provides a step-by-step method to solve it through coefficient comparison.

Introduction to Polynomial Equations

A polynomial equation is an equation that involves only non-negative integer powers of a variable. The general form of a polynomial equation is a_{n}x^{n} a_{n-1}x^{n-1} ... a_{1}x a_{0} 0

This article will walk through the process of solving the following polynomial equation:

ax^{3} - 1x - 2 bx^{2} - 1^{2} cx^{3} - 3^{3} equiv x^{3} 3x^{2} 3x 1

Step-by-Step Solution

To solve the equation, we will follow a systematic approach. First, we will expand both sides separately, and then equate the coefficients of corresponding powers of x.

Step 1: Expand the Right Side

The right side of the equation is ({x}^{3} 3{x}^{2} 3x 1). Expanding it yields:

({x}^{3} 3{x}^{2} 3x 1)

Step 2: Expand the Left Side

Next, we will expand each term on the left side:

Term 1: (ax(x-1)(x-2))

(ax(x-1)(x-2) ax^3 - a3x^2 2ax)
Simplify to: (ax^3 - 3ax^2 2ax)

Term 2: (b(x-1)^2 bx^2 - 2bx b)

Term 3: (c(x-3)^3 cx^3 - 9cx^2 27cx - 27c)

Step 3: Combine the Left Side

Now, we combine all the terms from the left side:

(cx^3 (a b - 9c)x^2 (2a - 2b 27c)x (2a b - 27c))

Step 4: Set the Coefficients Equal

We equate the coefficients of corresponding powers of x from both sides:

For ({x}^{3}) term:
(c 1) For ({x}^{2}) term:
(a b - 9c 9) For ({x}^{1}) term:
(-3a - 2b 27c 27) For constant term:
(2a b - 27c 27)

Step 5: Solve the System of Equations

We have the following system of equations based on the coefficients:

(c 1)

(a b - 9 9 Rightarrow a b 18)

(-3a - 2b 27 27 Rightarrow -3a - 2b 0 Rightarrow 3a 2b 0)

(2a b - 27 27 Rightarrow 2a b 54)

To solve for (a) and (b), we start with the second and third equations. From (3a 2b 0), we can express (b) in terms of (a):

(b -frac{3}{2}a)

Substitute (b) into (a b 18):

(a - frac{3}{2}a 18 Rightarrow -frac{1}{2}a 18 Rightarrow a -36)

Substitute (a -36) into (2a b 54):

(2(-36) b 54 Rightarrow -72 b 54 Rightarrow b 126)

Add (b 126) and (c 1):