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Solving Cubic Equations




 
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More Algebra Lessons, More Algebra Worksheets, More Algebra Games

What is the Remainder Theorem?
If a polynomial, f(x), is divided by x - k, the remainder is equal to f(k).

What is the Factor Theorem?
x - k is a factor of the polynomial f(x) if and only if f(k) = 0

How to solve cubic equations using the Factor Theorem?
In these lessons, 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.

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

Step 1: 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.

Step 2: Find the other roots 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



How to use the Factor Theorem to factor polynomials?
Examples:
1) Factor P(x) = 3x3 − x2 − 10x + 8
2) Factor P(x) = 2x3 − 9x2 + x + 12 What are The Remainder Theorem and the Factor Theorem?
How to use the Theorems to find the linear factorization of a polynomial?

Example: Factor F(x) = 2x3 − 3x2 − 5x + 6


 
How to use the Factor Theorem to solve a cubic equation?
If f(x) is a polynomial and f(p) = 0 then x - p is a factor of f(x)
Example: Solve the equation 2x3 −5x2 − 10 = 23x How to solve Cubic Equations using the Factor theorem and Long Division?
Example: Find the roots of the cubic equation 2x3 − 6x2 + 7x − 1 = 0

How to solve Cubic Equations using the Factor theorem and Synthetic Division?
Example: Show that x + 3 is a factor of x3 − 19x − 30 = 0. Then find the remaining factors of f(x) How to solve cubic equation problems?
Example: 3x3 −4x2 − 17x = x3 + 3x2 − 10
Step 1: Set one side of equation equal to 0.
Step 2: Collect like terms.
Step 3: Factorize using the Factor Theorem and Long Division

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