Polynomial Operations and Zeros

The video tutorial explains the complex conjugates theorem, which states that if a complex number a + bi is a zero of a polynomial, then its conjugate a - bi is also a zero. The tutorial demonstrates creating a polynomial equation with zeros of 4, 3 - i, and 3 + i. It walks through the process of rewriting the zeros in equation form, multiplying them, and simplifying the expression to derive the final polynomial equation: X^3 - 10X^2 + 24X - 30.

What does the complex conjugates theorem state about the zeros of a polynomial?

• If a + bi is a zero, then b - ai is also a zero.

• If a + bi is a zero, then a - bi is not a zero.

• If a + bi is a zero, then a + bi is the only zero.

• If a + bi is a zero, then a - bi is also a zero.

What is the purpose of using the complex conjugates theorem in this context?

• To ensure the polynomial has real coefficients.

• To ensure all zeros are real numbers.

• To find the product of the zeros.

• To find the sum of the zeros.

Which of the following is NOT a zero of the polynomial given the zeros 4, 3 - i, and 3 + i?

• 4

• 3 - i

• 3 + i

• 5

What is the imaginary part of the zero 3 + i?

• i

• -i

• -3

• 3

What is the real part of the zero 3 - i?

• i

• -i

• -3

• 3

How do you express the zero 3 - i as a factor in a polynomial equation?

• X + 3 + i

• X - 3 - i

• X + 3 - i

• X - 3 + i

Which of the following is a correct factorization of the polynomial with zeros 4, 3 - i, and 3 + i?

• (X - 4)(X - 3 - i)(X - 3 + i)

• (X + 4)(X + 3 - i)(X + 3 + i)

• (X + 4)(X - 3 - i)(X - 3 + i)

• (X - 4)(X + 3 - i)(X + 3 + i)

What is the first step in simplifying the polynomial with factors (X - 4), (X - 3 - i), and (X - 3 + i)?

• Subtract the factors.

• Divide the factors.

• Multiply the factors together.

• Add the factors together.

What is the result of multiplying (X - 3 - i) and (X - 3 + i)?

• X^2 - 9

• X^2 - 6X + 9

• X^2 - 6X + 10

• X^2 + 6X + 10

What is the result of simplifying (X - 3)^2 - i^2?

• X^2 - 6X + 9

• X^2 - 6X + 10

• X^2 - 9

• X^2 + 6X + 10