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Solving Cubic Equations




In this lesson, we will consider how to solve cubic equations of the form
px3 + qx2 + rx + s = 0 where p, q, r and s are constants by using the Factor Theorem and Synthetic Division.

Related Topics:
More Algebra Lessons

More Algebra Worksheets

More Algebra Games

Example:

Find the roots of f(x) = 2x3 + 3x2 – 11x – 6 = 0, given that it has at least one integer root.

Solution:

Since the constant in the given equation is a 6, we know that the integer root must be a factor of 6. The possible values are

We can use the factor theorem to test the possible values by trial and error.

f(1) = 2 + 3 – 11 – 6 ≠ 0

f(–1) = –2 + 3 + 11 – 6 ≠ 0

f(2) = 16 + 12 – 22 – 6 = 0

We find that the integer root is 2. Now we need to find the other roots. We can do that either by inspection or by synthetic division.

2x3 + 3x2 – 11x – 6
= (x – 2)(ax2 + bx + c)
= (x – 2)(2x2 + bx + 3)
= (x – 2)(2x2 + 7x + 3)
= (x – 2)(2x + 1)(x +3)

So, the roots are

Example:

Solve the cubic equation x3 – 7x2 + 4x + 12 = 0

Solution:

Let f(x) = x3 – 7x2 + 4x + 12

The possible values are

We find that f(–1) = –1 – 7 – 4 + 12 = 0

So, (x + 1) is a factor of f(x)

x3 – 7x2 + 4x + 12
= (x + 1)(x2 – 8x + 12)
= (x + 1)(x – 2)(x – 6)

So, the roots are –1, 2, 6



 

Videos

This video demonstrates how to use the Factor Theorem to factor polynomials.
1) Factor P(x) = 3x3 − x2 − 19x + 8
1) Factor P(x) = 2x3 − 9x2 + x + 12

The Remainder Theorem and the Factor Theorem :
What the theorems are and how they can be used to find the linear factorization of a polynomial.
Factor F(x) = 2x3 − 3x2 − 5x + 6




 

Using the Factor Theorem to solve a cubic equation
Solve the equation 2x3 −5x2 − 10 = 23x

How to solve a cubic equation



 

Solving Cubic Equations using the Factor theorem and Long Division
Solve the cubic equation 2x3 − 6x2 + 7x − 1 = 0


Solving a Cubic Equation
3x3 −4x2 − 17x = x3 + 3x2 − 10






You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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