Determine the equation of the hyperbola satisfying the given conditions. Write the answer in the form
Ax2 − By2 = C
or in the form
Ay2 − Bx2 = C.
Foci (0, ±25); vertices (0, ±7)

Determine the equation of the hyperbola satisfying the given conditions. Write the answer in the form
Ax2 − By2 = C
or in the form
Ay2 − Bx2 = C.
Asymptotes y = ±x; foci (0, ±3)

Use the remainder theorem (as in Example 3) to evaluate f(x) for the given value of x.
f(x) = x7 − 8x6 + 7x4 + 4; x = −3
f(-3) =

Find a polynomial equation
f(x) = 0
satisfying the given conditions. (If not possible, enter IMPOSSIBLE.)
Degree 4; the coefficients are integers; 1/2 is a root of multiplicity two;
3x2 − 4x − 2 is a factor of f(x)
Express the polynomial
f(x)
in the form
anxn + an − 1xn − 1 + + a1x + a0.

Find a quadratic equation with the given roots. Write your answer in the form
Ax2 + Bx + C = 0.
Suggestion: Make use of the table given below.
r1 = 2 − 4i, r2 = 2 + 4i

List the possibilities for rational roots. (Enter your answers as a comma-separated list.)
18x4 − 8x3 + x2 − 4 = 0

Find the rational roots of the equation, and then solve the equation. (Use the rational roots theorem and the upper and lower bound theorem, as in Example 2. Enter your answers as a comma-separated list.)

2x3 − 3x2 − 9x + 5 = 0
An equation is given, followed by one or more roots of the equation. Determine the remaining roots. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)