MIT Algebra Worksheet

Review Problems1. Write each complex number in standard form a +bi.
(a) (1 – 3i) + (7 — 4i)
(b)
(5 – 3i) – (2 + 7i)
–
(c) (5 + 3i) · (2 + 7i)
(e)
1
(d)
3 + 4i
(g) (2+i)3
1+ 4i
2 – 3i
(f)
-2 V-54
4
2. Find all solutions (real and complex)
(a) x2 + 25 = 0
(b) x2 – 4x = -8
(C) 3×2 + 5 = -41
3. Find all real solutions for x and simplify your answers:
х
(a) 2×2 = 6 – x
=
1
2x + 1
(b) +
X – 2
x + 3
x2 + x – 6
(d) x4
8×2 – 9= 0
(c) x2 + 2x – 2 = 0
–
(e) –5 = x + V x + 17
(g) (2x – 1)1/3 = 2
(i) 7 – 4(x + 2) = 3x + 5
=
(f) 23 + 2×2 – 3x — 6 = 0
(h) x6 + 3×3 – 40 = 0)
(i) 4x + 71 = 19
=
=
4. Simplify each of the following. Write any polynomials in standard form.
1
2
2
(a)
(b)
(c) 5×2 – X – 3(x − 2)2
—
2
+1
१२ |
х
x3 + 8
x² – 4
(d) 5(x + 1)3
(e)
(f)
x2
x2 – X – 6
X + 2
x6
(g)
(x – 5) + 7
(x – 5)(x + 2)
5. Factor completely:
(a) 9×2 – 25y2
(d) y3 + 2y2 – 9y – 18
(g) 8×2 – 14x 15
(b) x4 + 3×2 – 4
(c) a} – 2763
(e) 4a² (3x – 1) – 9(3x – 1) (f) 27×3 + 1
=
2
6. Let f(x) = 174 and g(x) = x2 – 3. Find and simplify:
(a) (fog)(x) (b) (gof)(2) (c) g(x+h)
(e) (909) (2) (f) -(0)
(g) f(-1)
(0) (-9) ()
(d) f-1(x)
(h) (f.g) (2)
7. Consider the quadratic function f(x) = –2×2 – 8x + 10.
(a) Write in form y =
= a(x – h)2 + k. (b) Find the vertex.
(c) Find the y-intercept.
(d) Find any x-intercept(s).
8. Solve each of the following. Write your answer in interval notation.
3.0
(a) -7