Question

Use the given zeros to find the remaining zeros of the function. 

F(x)=x^3-8x^2+25x-200; zeros: 5 i

Enter the remaining zeros of f.

Answer 1

f (x ) = x³ - 8x²  + 25x - 200

One of the real zero is 5i.

The polynomial function is x³ - 8x²  + 25x - 200 and the root is 5i.

Use synthatic division to find

Write the terms of the dividend so that the degrees of the terms are in descending order. Then write just the coefficients as shown in below.

Step 2: Write the constant r of the divisor (x - r) to the left. In this case, r = 5i. Bring the first coefficient 1, down.

Step 3: Multiply the first coefficient by r :5i (1). Write the product under the second coefficient, - 8 and add : -8 + 5i.

Step - 4:

Multiply the sum, - 8 + 5i, by r : 5i(- 8 + 5i) = - 40i - 25.

Write the product under the next coefficient, 25 and add : 25 + (- 40i - 25) = - 40i.

Step -  5:

Multiply the sum,- 40i, by r : 5i(- 40i) = 200 .

Write the product under the next coefficient, -200 and add : - 200 + (200) = 0. The remainder is 0.

The numbers along the bottom row are the coefficients of the quotient. Start with the power of x  that is one less than the degree of the dividend. Thus, the quotient is

The resulting equation is and solve for x  by using factors method.

The function f (x ) = x³ - 8x² + 25x - 200 have one real zero at x  = 8 and two imaginary zeros at x  = 5i and x  = - 5i.

Therefore, remaining zeros are 8 and - 5i.