[1] State the degree of the polynomial: x³ - x² + x⁴

[2] Remove parentheses and combine like terms: -4m² + 3n² - 5n - [3m² - 5n² + 2n - (3m² - 4n²)]

[3] Factor out the largest common factor: 9x²y²z⁴ - 18x³y²z

[4] Factor out the largest common factor, and then combine like terms: 4(3-x)² - (3-x)³ + 3(3-x)

[5] Factor: 27x³ - 64y³

[6] Factor, i.e., write as A times B for some A & B: (m-p)² + 4(m-p) + 4

[7] Simplify by writing as one term (it is 3 terms, now): 3 - -

= 3 - 2 - 6 = -5

[8] Solve for x: - = 1 : square both sides to get

()² - 2^. + ()² = 1²

(x + 2) - 2^. + (x - 3) = 1

2x - 2 = 2^.

x - 1 = ^. : now square both sides again to get

x² - 2x + 1 = (x + 2)(x - 3) = x² -x - 6 : the red terms cancel

-2x + 1 = -x - 6

x = 7

[9] Find all values of x for which 10x² + x - 21 = 0

    Two possible methods:

[A] Use quadratic formula with a=10, b=1, and c=-21, to get x =

-1 ± [12 - 4(10)(-21)]½ 2(10)

    [B] Factor the given left side, to get (2x + 3)(5x - 7) = 0,
     In either above case, we find either x = -3/2 , or x = +7/5

[10] Solve for x: log(4x) = log(2) + log(x-3): combine right side as

A couple of students correctly noticed that x = -3 will make numbers in the original equation into complex numbers, and in some algebra courses, such situations are interpreted as if there were no solution (actually, no "real" solution)