Synthetic Division

Question

Use synthetic division to solve:

x⁵ - 4x⁴ - 2x³ + 4x² + x = 0

Answer 1

The first step you can do is factor out the GCF, which is x. The result is:

x(x⁴ - 4x³ - 2x² + 4x + 1) This gives the first zero, x=0

Now, we can use synthetic division. Possible factors would be 1 or -1, we can try both of those.

1 | 1 -4 -2 4 1

1 -3 -5 -1

1 -2 -5 -1 0 The zero in the last position means a remainder of 0, which confirms the zero x=1

We can use synthetic division again with the result as the next step and try x=-1

-1 | 1 -2 -5 -1

-1 4 1

1 -4 -1 0 Once again, zero as the remainder confirms that we have another zero, x=-1

The polynomial that remains is x²-4x-1 This can be solved using the quadratic formula.

a=1 b=-4 c=-1

[-(-4)±√(-4)²-5(1)(-4)]/[2(1)] Simplify

[4±√20]/2 Simplify

4/2±2√5/2 Simplify

2±√5

In summary, the 5 solutions are:

x=0,1,-1,2±√5

Answer 2