p(x)=5x^3-4x^2+20-16
use synthetic division to show that x=2i is a zero of p(x)
Your function, which I will assume was to be
p(x) = 5x^3 - 4x^2 + 20x - 16 , factors very nicely by grouping
p(x) = 5x^3 - 4x^2 + 20x - 16
= x^2(5x-4) + 4(5x-4)
= (5x-4)(x^2 + 4)
so the zeros are
x = 4/5 and x = ± 2i
Using synthetic division for this expression is totally out of place.
lim x�¨�‡ 2x 3 +5x 2 −x 3 =
To use synthetic division to show that x = 2i is a zero of p(x), we need to set up the synthetic division table to divide the polynomial p(x) by (x - 2i).
First, let's rewrite the polynomial p(x) in descending order of powers:
p(x) = 5x^3 - 4x^2 - 16 + 20
Now, we set up the synthetic division table like this:
2i │ 5 -4 0 20 - 16
Let's perform the synthetic division step-by-step:
Step 1: Bring down the coefficient of the highest power term, which is 5:
2i │ 5 -4 0 20 - 16
──────────────
5
Step 2: Multiply 2i by the number obtained in the previous step (5), and write the result under the next coefficient:
2i │ 5 -4 0 20 - 16
──────────────
5
10i
Step 3: Add the respective coefficients in the first and second columns:
2i │ 5 -4 0 20 - 16
──────────────
5 6i
Step 4: Repeat steps 2-3 until all coefficients are included:
2i │ 5 -4 0 20 - 16
──────────────
5 6i 12 - 20i
Step 5: The last number in the right column gives us the remainder. In this case, the remainder is -20i.
Since the remainder is not zero, x = 2i is not a zero of p(x).