a.
The mistake in this claim is that its complex conjugate is also a root of the quadratic polynomial equation.
Since the student claims the polynomial has only one imaginary fooot of -9i, this is impossible because, a polynomial with real coefficients that has a complex root also has its conjugate as a root of the quadratic polynomial equation.
So, the mistake in this claim is that its complex conjugate is also a root of the quadratic polynomial equation.
b.
One possible factored polynomial that has the correct roots is P(x) = (x + 9i)(x - 9i)
Since -9i is a root, its conjugate +9i is also a root.
So, the factors of the quadratic polynomial are x - (-9i) and x - 9i. Which are x + 9i and x - 9i.
To obtain the factored polynomial, we multiply its factors together.
So, P(x) = (x + 9i)(x - 9i)
So, one possible factored polynomial that has the correct roots is P(x) = (x + 9i)(x - 9i).
c.
The factored polynomial in standard form is x² + 81
To write the polynomial in standard form, we expand the brackets.
So,
P(x) = (x + 9i)(x - 9i)
= x² - (9i)² (difference of two squares)
= x² - 81i²
= x² - 81 × (-1)
= x² + 81
So, the factored polynomial in standard form is x² + 81.
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