How to solve third degree polynomial and give examples
To solve a third degree polynomial, you can use either factoring, the rational root theorem, or synthetic division. Here are the steps to solve a third degree polynomial using factoring:
1. Write the polynomial in standard form: \(ax^3 + bx^2 + cx + d\)
2. Look for common factors among the coefficients and factor them out if possible.
3. Try to factor the polynomial into a product of binomials or trinomials. You may need to use techniques like grouping, difference of squares, or completing the square to factor the polynomial.
4. Once you have factored the polynomial completely, set each factor equal to zero and solve for the roots of the polynomial.
5. Check your solutions by substituting them back into the original polynomial to make sure they are correct.
For example, let's solve the polynomial \(2x^3 - 11x^2 + 10x\) using factoring:
Step 1: Write the polynomial in standard form: \(2x^3 - 11x^2 + 10x\)
Step 2: There are no common factors to factor out.
Step 3: Factor by grouping: \(2x^3 - 11x^2 + 10x = x(2x^2 - 11x + 10) = x(2x - 5)(x - 2)\)
Step 4: Set each factor equal to zero and solve for x:
\(x = 0\)
\(2x - 5 = 0 \Rightarrow x = 5/2\)
\(x - 2 = 0 \Rightarrow x = 2\)
So, the solutions to the polynomial \(2x^3 - 11x^2 + 10x\) are \(x = 0\), \(x = 5/2\), and \(x = 2\).