The polynomial x³ - 4x² - 7x + 10 has x=1 as one of its roots. What is the product of the other two roots?

Since $x = 1$ is a root, divide the cubic by $x - 1$. The quotient is $x^2 - 3x - 10$, which you can confirm by multiplying: $(x-1)(x^2 - 3x - 10) = x^3 - 4x^2 - 7x + 10$. The other two roots solve $x^2 - 3x - 10 = 0$, which factors as $(x - 5)(x + 2) = 0$, giving roots 5 and $-2$. Their product is $5 \times (-2) = -10$. A common wrong answer is 10, from dropping the sign. Another is 3, the sum of the other roots. A third is $-5$, which is just one factor. Vieta's formula gives the product as the constant term of the quotient.