Expanding (z - α)(z - β) = 0 yields which quadratic?

Expanding (z - α)(z - β) = 0 yields which quadratic?

• A. z^2 - (αβ) z + α + β = 0
• B. z^2 - (α + β) z + αβ = 0
• C. z^2 + (α + β) z - αβ = 0
• D. z^2 - (α + β) z - αβ = 0

Answer: (B)

When you expand two linear factors, the z^2 term appears, the cross terms combine to give a z-term with the negative of the sum of the constants, and the constant term is the product of the constants. So, (z - α)(z - β) expands to z^2 - βz - αz + αβ, which is z^2 - (α + β)z + αβ. Setting this equal to zero gives z^2 - (α + β)z + αβ = 0. This matches the form where the z-term coefficient is minus the sum α + β and the constant term is αβ, which is why this is the correct quadratic. The other forms would have incorrect signs for either the z-term or the constant term.

When you expand two linear factors, the z^2 term appears, the cross terms combine to give a z-term with the negative of the sum of the constants, and the constant term is the product of the constants. So, (z - α)(z - β) expands to z^2 - βz - αz + αβ, which is z^2 - (α + β)z + αβ. Setting this equal to zero gives z^2 - (α + β)z + αβ = 0. This matches the form where the z-term coefficient is minus the sum α + β and the constant term is αβ, which is why this is the correct quadratic. The other forms would have incorrect signs for either the z-term or the constant term.