Cung HA2 S2 Final Review

Polynomial Equations

100

Using the Rational Root Theorem, list the possible rational roots of:

f(x) = -3x³+x⁵+2x²-6

 Possible Rational Roots:

±1, ±2, ±3, ±6

200

What is the remainder of:

(5𝑥⁴ − 2𝑥³ − 7𝑥² − 39)÷(𝑥² + 2𝑥 − 4)

-122x + 109

300

Divide (3𝑥⁴ − 17𝑥³ + 285𝑥 − 51𝑥² + 100) ÷ (𝑥−5)²

3𝑥² + 13𝑥 + 4

400

Find all possible k such that -2 is a root of 3x³ + kx² + k²x + 40.

4, -2

500

Find a fourth degree polynomial with real coefficients and roots at 3, -3, -3+i. The polynomial crosses (-2, -20). Write in expanded form.

2x⁴ + 12x³ +2x² - 108x - 180

Conics

100

Write the equation for a parabola with a focus at (2, -1) and directrix at x = 6.

-1/8(y + 1)² = x - 4

200

Solve:

x² + y² = 13

xy + 6 = 0

(3, -2), (-3, 2), (2, -3), (-2, 3)

300

Write in general form. Find the equation of a hyperbola with one vertex at (-1, -6) and asymptotes at:

(y - 2) = ±(2/3)(x+1)

(y-2)²/64 - (x+1)²/144 = 1

400

Find the equation of a circle that passes through the following three points: (-6, 0), (-5, -1), and (1, -1)

(x+2)² + (y -3)² = 25

x² + y² + 4x - 6y - 12 = 0

500

The arch of a bridge is in the form of half an ellipse, with the major axis horizontal. The span of the bridge is 12 meters and the height of the arch above the water is 4 meters. How high above the water is the arch at a point on the water 1 meters from one of the ends of the arch? Give the exact answer.

2sqrt(11)/3 meters

Logarithms

100

Fully expand: log(ab²/c³)

log a + 2 log b - 3 log c

200

Simplify log_1/9(243)

-5/2 or -2.5

300

A substance has a half-life of 8 years. How much of the original substance is present after 4 years? Give the exact answer. 

1/sqrt(2)

400

Solve for x:

log₄x - log₄(x-1) = 1/2

2

500

2 + log(1+x)^1/2 + 3log(1-x)^1/2 = log(1-x²)^1/2

x = 99/100

Trigonometry

100

Convert π° (pi degrees) to radians and find the highest negative coterminal in radians

π² / 180 with coterminal: π² / 180 - 2π

200

Write an equation for the graph shown such that there is no phase shift:

-2sin(πx)+1

300

Rewrite cos(8π/7) so that it is in terms of sin(x) where x is the smallest possible positive value.

sin(19π/14)  or -sin(5π/14)

400

Write the equation of the function if the two points given (0.5, 1) and (1.5, -2) are NOT on the midline. There are asymptotes at x = 0 and x = 2. Use tangent.

-1.5tan(π/2 (x-1) ) - 0.5

or

-1.5tan(π/2 (x+odd integer) ) - 0.5

500

Quadrilateral QUAD has the following known side lengths: QU = 16, UA = 8, AD = 10

It also has angles U = 55° and A = 101°

What is the area of the QUAD?

Area is approximately 59.16

Sequences and Series

100

Find the sum of:

7 + 1/5 + 14 + 1/10 + 28 + 1/20 + ... + 224

441+31/80

200

Solve for x = log(900+90+9+.9+.09+...)

3

300

What is the product of e⁵, e⁶, e⁷, e⁸,..., e¹⁰⁵?

e⁵⁵⁵⁵

400

Find the sum

-1-sqrt(3)/2

500

Rewrite the following so that the series starts at k = 12

or (k-6)² - 10