Question

Write a polynomial function of least degree with rational coefficients so that P (x)=0 has the given root. 3-5i

P(x)=x2-◻️x+◻️

Answer 1

To write a polynomial function of least degree with rational coefficients with the root 3-5i, we need to also include the conjugate of the root. The polynomial function is x^2 - 6x + 34.

Step-by-step explanation:

To write a polynomial function of least degree with rational coefficients that has the root 3-5i, we need to also include the conjugate of the root, which is 3+5i. This is because complex roots always come in conjugate pairs. So, the polynomial function would be:

P(x) = (x - (3 - 5i))(x - (3 + 5i))

To expand this, we can use the distributive property:

P(x) = (x - 3 + 5i)(x - 3 - 5i)

Expanding the expressions:

P(x) = (x - 3)(x - 3) + (x - 3)(-5i) + (5i)(x - 3) + (5i)(-5i)

Simplifying:

P(x) = (x^2 - 6x + 9) + (-5ix + 15i) + (5ix - 15i) + (25)

Combining like terms:

P(x) = x^2 - 6x + 9 - 5ix + 15i + 5ix - 15i + 25

Further simplifying:

P(x) = x^2 - 6x + 34

Answer 2

will not be anything