Use the given zero to find the remaining zeros of the function.
(hx)= 3x^4+5x^3+25x^2+45x-18; zero:-3i
labe
complex roots always come in conjugate pairs, so we have
x = -3i and x = 3i
two factors are (x + 3i) and (x- 3i)
(x+3i)(x-3i)
= x^2 - 9i^2 = x^2 + 9
Using long algebraic division, you should get
( 3x^4+5x^3+25x^2+45x-18) ÷ (x^2+9)
= 3x^2 + 5x + 2
the quadratic formula can be used to find the other two roots.
btw, it factors
To find the remaining zeros of the function, we can use a method called synthetic division. Synthetic division is a simplified form of long division specifically used for dividing polynomials by binomials of the form (x - c), where "c" is the zero we know.
In this case, the given zero is -3i. Since it is an imaginary number, complex conjugates always appear in pairs. So, the complex conjugate of -3i is 3i.
To find the remaining zeros, we will use synthetic division. Let's start with the complex conjugate zero, 3i.
Step 1: Set up the synthetic division table:
| 3i
---------------------
-3i | 3 5 25 45 -18
Step 2: Use the first coefficient (3) as the first entry in the second row.
| 3i
---------------------
-3i | 3 5 25 45 -18
Step 3: Multiply the divisor (3i) by the result in the second row (3) and write the product below the next coefficient (5). Then add the values in the second row:
| 3i
---------------------
-3i | 3 5 25 45 -18
9i
| 3i
---------------------
-3i | 3 5 25 45 -18
9i 14i
Step 4: Repeat step 3 for the next coefficient:
| 3i
---------------------
-3i | 3 5 25 45 -18
9i 14i
42
| 3i
---------------------
-3i | 3 5 25 45 -18
9i 14i
42 45i
Step 5: Repeat step 3 for the remaining coefficients:
| 3i
---------------------
-3i | 3 5 25 45 -18
9i 14i
42 45i
-135
| 3i
---------------------
-3i | 3 5 25 45 -18
9i 14i
42 45i
-135 48i
Step 6: The last entry in the second row, -135, is the remainder. If the remainder is zero, it means that 3i is a zero of the function. In this case, the remainder is non-zero, so 3i is not a zero.
Step 7: The coefficients in the second row, excluding the last entry, are the coefficients of a lower-degree polynomial. In this case, the polynomial we have now is 3x^3 + 9ix^2 + 14ix + 42.
To find the remaining zeros, we repeat the process using this new polynomial as our starting point. However, since the coefficients involve complex numbers, the remaining zeros can be complex or real.
Repeat the steps starting from Step 1, but use the polynomial 3x^3 + 9ix^2 + 14ix + 42 instead of the original function.
By repeating the synthetic division process using the new polynomial, you can find the remaining zeros of the function.