Step-By-Step Guide to Solving Polynomials

Polynomials are algebraic expressions that involve variables, coefficients, and exponents. They are commonly encountered in mathematics and are used in various fields such as engineering, physics, and even computer science. Solving polynomials involves finding the values of the variables that make the expression equal to zero. In this step-by-step guide, we will walk through the process of solving polynomials.

Step 1: Identify the Polynomial

The first step is to identify the polynomial you want to solve. Polynomials usually have the form ax^n + bx^(n-1) + cx^(n-2) + … + zx^0, where ‘a’ through ‘z’ represent coefficients, ‘x’ is the variable, and ‘n’ is the highest exponent in the polynomial.

For example, let’s consider the polynomial 3x^3 + 5x^2 – 2x – 4.

Step 2: Factorization (if possible)

If the polynomial can be factored, it will help simplify the solving process. The goal is to rewrite the polynomial as a product of its factors. To do this, you need to identify any common factors and apply factoring techniques like grouping, difference of squares, or trinomial factoring.

Using our example polynomial, 3x^3 + 5x^2 – 2x – 4, we can start by grouping pairs of terms:

(3x^3 + 5x^2) + (-2x – 4)

Now, factor out common terms from each group:

x^2(3x + 5) – 2(1x + 2)

This polynomial cannot be further factored, so we move on to the next step.

Step 3: Set the Polynomial Equal to Zero

Once the polynomial is factored or if it cannot be factored, set the polynomial equal to zero. This means that the sum of all terms in the polynomial would be equal to zero.

Using our example: x^2(3x + 5) – 2(1x + 2) = 0.

Step 4: Solve for the Variables

To solve for the variables, we will apply the zero product property. The zero product property states that if a product of two or more factors is equal to zero, then at least one of the factors must be zero.

Given our example equation: x^2(3x + 5) – 2(1x + 2) = 0.

We set each factor equal to zero:

x^2 = 0 or 3x + 5 = 0 or 1x + 2 = 0.

Solving each equation leads to:

x = 0, x = -5/3, x = -2.

Therefore, the polynomial 3x^3 + 5x^2 – 2x – 4 has three solutions: x = 0, x = -5/3, and x = -2.

Step 5: Check the Solutions

After obtaining the solutions, it is important to check if they are valid or not. To do this, substitute each solution back into the original polynomial equation and verify that it equals zero.

Using our example polynomial:

When x = 0, 3(0)^3 + 5(0)^2 – 2(0) – 4 = 0, which is true.

When x = -5/3, 3(-5/3)^3 + 5(-5/3)^2 – 2(-5/3) – 4 = 0, which is true.

When x = -2, 3(-2)^3 + 5(-2)^2 – 2(-2) – 4 = 0, which is true.

All solutions checked out, confirming their validity.

In conclusion, solving polynomials requires identifying the polynomial, factoring if possible, setting it equal to zero, solving for the variables, and finally checking the solutions. By following this step-by-step guide, you can solve polynomials and find their solutions efficiently.