Degree 5; leading coefficient 6; zeros:6 (multiplicity 2),2,4-i

Answer 1

If 6 is a zero, then (x - 6) is a factor of the polynomial. Since it has a multiplicity of 2, then (x - 6)² is a factor.

If 2 is a zero, then (x - 2) is a factor.

If 4 - i is a zero, there will ALWAYS be a conjugate zero that goes with it, therefore 4 + i will also be a zero. And, if 4 - i is a zero, then (x - (4 - i)) is a factor and if 4 + i is a zero, then (x - (4 + i)) is a factor.

So the polynomial can be y = (x - 6)²(x - 2)(x - (4 - i))(x - (4 + i)) but we are told the leading coefficient is 6 therefore it must be y = 6(x - 6)²(x - 2)(x - (4 - i))(x - (4 + i)) or y = 6(x - 6)²(x - 2)(x - 4 + i)(x - 4 - i)

We can multiply the last 2 terms together to get a better appearing result.

results in x² - 8x + 16 - i² but since i² = -1 then it becomes: x² - 8x + 16 - -1 = x² - 8x + 17

So the polynomial (in factored form) is y = 6(x - 6)²(x - 2)(x² - 8x + 17)

You can multiply it out further if it is required in the same way as above.

Answer 2

y=6(x-6)^2(x-2)(x-(4-i))(x+(4-i)).

Simplify as necessary.