Polynomial Equation
How to solve a polynomial equation by using the synthetic division: 1 example and its solution.
Examplex4 + 4x3 - 3x2 - 10x + 8 = 0
Factor the left side
by using the synthetic division.
Factor Theorem
f(x) = x4 + 4x3 - 3x2 - 10x + 8
Write the coefficients of the terms
in descending order:
1 4 -3 -10 8.
Draw an L shape form like this.
Pick a number
that seems to make the remainder 0.
1 seems to be good.
So write 1
on the left side of the form.
↓: 1 = 1
↗: 1⋅1 = 1
↓: 4 + 1 = 5
↗: 5⋅1 = 5
↓: -3 + 5 = 2
↗: 2⋅1 = 2
↓: -10 + 2 = -8
↗: -8⋅1 = -8
↓: 8 - 8 = 0
The remainder is 0.
So 1 is the right number.
1 5 2 -8 means
x3 + 5x2 + 2x - 8.
This seems to be factorable.
So do the synthetic division again.
Draw an L shape form like this.
Pick a number
that seems to make the remainder 0.
1 seems to be good.
So write 1
on the left side of the form.
↓: 1 = 1
↗: 1⋅1 = 1
↓: 5 + 1 = 6
↗: 6⋅1 = 6
↓: 2 + 6 = 8
↗: 8⋅1 = 8
↓: -8 + 8 = 0
The remainder is 0.
So 1 is the right number.
1 6 8 means
x2 + 6x + 8.
This seems to be factorable.
So do the synthetic division again.
Draw an L shape form like this.
Pick a number
that seems to make the remainder 0.
-2 seems to be good.
So write -2
on the left side of the form.
↓: 1 = 1
↗: 1⋅(-2) = -2
↓: 6 - 2 = 4
↗: 4⋅(-2) = -8
↓: 8 - 8 = 0
The remainder is 0.
So -2 is the right number.
1 4 means
x + 4.
(x + 4) is a binomial factor.
So stop finding the factors.
The left side numbers are
1, 1, and -2.
So write the factors
(x - 1)2(x + 2).
1 4 means
x + 4.
So write (x + 4).
Write = 0.
So the given equation is
(x - 1)2(x + 2)(x + 4) = 0.
Solve the equation
by finding the zeros of each factor.
The zero of (x - 1)2 is 1.
The zero of (x + 2) is -2.
The zero of (x + 4) is -4.
So x = 1, -2, -4.
So x = 1, -2, -4.