In example 1, the polynomial x^3-13x^2+43x-15 was reduced to the factors x-5 and x^2-8x+3. Find the zeros of the polynomial.

Answer 1

The zeros of a polynomial are the values of x that make the polynomial equal to zero.

In this case, the two factors we obtained from the reduction are (x-5) and (x^2-8x+3).

To find the zeros, we set each factor equal to zero and solve for x.

For the factor x-5:
x - 5 = 0
x = 5

For the factor x^2-8x+3, we can use the quadratic formula to solve for x.
The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a), given ax^2 + bx + c = 0.

In this case, a = 1, b = -8, and c = 3.
Substituting these values into the quadratic formula, we get:
x = (-(-8) ± √((-8)^2 - 4(1)(3))) / (2(1))
x = (8 ± √(64 - 12)) / 2
x = (8 ± √52) / 2
x = (8 ± 2√13) / 2
x = 4 ± √13

So, the zeros of the polynomial x^3-13x^2+43x-15 are x = 5, x = 4 + √13, and x = 4 - √13.