use the given zero to find the remaining zeros of the function.

h(x)=x^4-12x^3+36x^2+68x-525
zero:4-3i

complex zeros always come in conjugate pairs

so if 4-3i is a root, so is 4+3i

so (x - 4 -3i)(x - 4 + 3i) will be a factor
= x^2 - 4x + 3ix - 4x + 16 - 12i - 3ix + 12i - 9i^2)
= x^2 - 8x + 25

Using long algebraic division
(x^4 - 12x^3 + 36x^2 + 68x - 525) / (x^2 - 8x + 25)
= x^2 - 4x - 21

so for x^2 - 4x - 21 = 0
(x-7)(x+3) = 0
x = 7 or x = -3

roots are :
-3 , 7 , 4+3i , 4-3i
=

To find the remaining zeros of the function, we can use the fact that complex zeros come in conjugate pairs.

Given that one of the zeros is 4 - 3i, we know that the conjugate of this complex number is 4 + 3i.

To find the remaining zeros, we need to find the quadratic equation formed by these two conjugate zeros.

Using the fact that complex zeros appear in conjugate pairs, we can write the quadratic equation as:

(x - (4 - 3i))(x - (4 + 3i)) = 0

Expanding this equation:

(x - 4 + 3i)(x - 4 - 3i) = 0

Using the difference of squares:

(x - 4)^2 - (3i)^2 = 0

(x - 4)^2 + 9 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -8, and c = 25. Plugging these values into the quadratic formula:

x = (8 ± √((-8)^2 - 4(1)(25))) / (2(1))

x = (8 ± √(64 - 100)) / 2

x = (8 ± √(-36)) / 2

Since the discriminant is negative, we have imaginary solutions. Simplifying further:

x = (8 ± 6i) / 2

x = 4 ± 3i

Hence, the remaining zeros of the function are 4 + 3i and 4 - 3i.

To find the remaining zeros of the function given the zero 4-3i, we can use the property that complex zeros of polynomials with real coefficients always come in conjugate pairs.

Since 4-3i is a zero of the function, its conjugate, 4+3i, must also be a zero. Therefore, we now have two zeros: 4-3i and 4+3i.

To find the remaining zeros, we can use polynomial division or synthetic division. Starting with the function:

h(x) = x^4 - 12x^3 + 36x^2 + 68x - 525

We divide it by the quadratic factor formed by the known zeros:

(x - (4 - 3i))(x - (4 + 3i))

To do the division, we'll use synthetic division. The divisor is x^2 - 8x + 25, which is derived from (x - (4 - 3i))(x - (4 + 3i)).

1 - 8 | 36 68 -525
- 8 | -8 64
_____________________

1 - 8 -8 | 28 56 -461

The remainder is 28x + 56. The remaining quadratic that we obtained from the division is x^2 - 8x - 8.

Now, we can solve x^2 - 8x - 8 = 0 to find the remaining zeros. By factoring or using the quadratic formula:

x^2 - 8x - 8 = 0
(x - 4)(x + 2) = 0

Setting each factor equal to zero, we have:

x - 4 = 0 --> x = 4
x + 2 = 0 --> x = -2

Therefore, the remaining zeros of the function h(x) are 4 and -2.