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Exam Review Questions for College Algebra | MATH 143, Study notes of Algebra

Material Type: Notes; Professor: Kenny; Class: College Algebra; Subject: Mathematics; University: Boise State University; Term: Spring 2009;

Typology: Study notes

Pre 2010

Uploaded on 08/19/2009

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Math 143/7 Some Chapter 3 Review - No calculators bkenny.spring2009
1. Divide the following, writing (a) in the P(x) = D(x)Q(x) + R(x) form,
and (b) in the P(x)
D(x)=Q(x) + R(x)
D(x)form.
a) 5x43x2+ 2
x23x+ 5 b) x5+ 2
x+ 3
2. Find all the zeros of 3x35x216x+ 12 = 0, given 2 is a zero.
3. Find all real solutions of
a) 3x514x414x3+ 36x2+ 43x+ 10 = 0 b) 3x4+ 5x2+ 2 = 0
4. Evaluate the expression, writing the answer in a+bi form.
a) (3 + 5i)(4 2i) b) 3 + 5i
12ic) (12 3)(3 + 4) d) 749
28
5. Find all solutions (both real and complex) of the equations given.
a) x364 = 0 b) x41 = 0 c) x6+ 7x38 = 0
6. Find the complete factorization of P(x) = x42x32x22x3.
7. Given each of the following functions, find the intercepts, the asymptotes, and graph
the function.
a) y=4x8
(x4)(x+ 1) b) y=3x2+ 6
x22x3
c) y=x22x+ 1
x2+ 2x+ 1 d) y=x29
2x2+ 1 e) y=3x+ 6
x2+ 2x8
8. Does there exist a polynomial of degree 6 with integer coefficients that has zeros
i, 2i, 3i, 4i? If so, find it. If not, explain why.
9. What is the remainder when x101 x4+ 2 is divided by x+ 1?
10. Find the coordinates of all points of intersection of the graphs of y=x4+x2+ 24x
and y= 6x3+ 20. (Do not do this by graphing.)
11. Graph f(x) = x4+x32x2, showing the correct xintercepts and end behavior.
12. Solve, using exact values. x5x3= 2x
13. Find the degree, the end behavior, and yintercept of
f(x) = ax8+bx4+cx2+dx +e, a < 0
14. Find bso x+ 2 is factor of f(x)f(x) = x2bx + 5
15. f(x) = ax2+bx +c
dx3+ex +fhas at most
a) how many xintercepts? b) how many vertical asymptotes?
16. Graph f(x) = a(xb)(xc)2(xd)2, a, b < 0, c, d > 0, c < d
pf2

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Math 143/7 Some Chapter 3 Review - No calculators bkenny.spring

  1. Divide the following, writing (a) in the P (x) = D(x)Q(x) + R(x) form, and (b) in the P D^ ((xx)) = Q(x) + (^) DR((xx)) form.

a)^5 x

(^4) − 3 x (^2) + 2 x^2 − 3 x + 5 b)^

x^5 + 2 x + 3

  1. Find all the zeros of 3x^3 − 5 x^2 − 16 x + 12 = 0, given −2 is a zero.
  2. Find all real solutions of a) 3x^5 − 14 x^4 − 14 x^3 + 36x^2 + 43x + 10 = 0 b) 3x^4 + 5x^2 + 2 = 0
  3. Evaluate the expression, writing the answer in a + bi form. a) (3 + 5i)(4 − 2 i) b) 3 + 5 1 − 2 ii c) (√ 12 − √−3)(3 + √−4) d)
  1. Find all solutions (both real and complex) of the equations given. a) x^3 − 64 = 0 b) x^4 − 1 = 0 c) x^6 + 7x^3 − 8 = 0
  2. Find the complete factorization of P (x) = x^4 − 2 x^3 − 2 x^2 − 2 x − 3.
  3. Given each of the following functions, find the intercepts, the asymptotes, and graph the function. a) y = (^) (x −^4 x4)(^ −x^8 + 1) b) y = 3 x

x^2 − 2 x − 3 c) y = x

(^2) − 2 x + 1 x^2 + 2x + 1 d)^ y^ =^

x^2 − 9 2 x^2 + 1 e)^ y^ =^

3 x + 6 x^2 + 2x − 8

  1. Does there exist a polynomial of degree 6 with integer coefficients that has zeros i, 2 i, 3 i, 4 i? If so, find it. If not, explain why.
  2. What is the remainder when x^101 − x^4 + 2 is divided by x + 1?
  3. Find the coordinates of all points of intersection of the graphs of y = x^4 + x^2 + 24x and y = 6x^3 + 20. (Do not do this by graphing.)
  4. Graph f (x) = x^4 + x^3 − 2 x^2 , showing the correct x−intercepts and end behavior.
  5. Solve, using exact values. x^5 − x^3 = 2x
  6. Find the degree, the end behavior, and y−intercept of f (x) = ax^8 + bx^4 + cx^2 + dx + e, a < 0
  7. Find b so x + 2 is factor of f (x) f (x) = x^2 − bx + 5
  8. f (x) = ax

(^2) + bx + c dx^3 + ex + f has at most a) how many x−intercepts? b) how many vertical asymptotes?

  1. Graph f (x) = a(x − b)(x − c)^2 (x − d)^2 , a, b < 0 , c, d > 0 , c < d

Answers:

  1. a) 5x^4 − 3 x^2 + 2 = (x^2 − 3 x + 5)(5x^2 + 15x + 17) + − 24 x − 83 b) x

x + 3 =^ x

(^4) − 3 x (^3) + 9x (^2) − 27 x + 81 − 241 x + 3

  1. x = − 2 , 3 , (^23)
  2. a) x = −1(mult2), 2 , 5 , −^13 b) 3x^4 + 5x^2 + 2 = (3x^2 + 2)(x^2 + 1) There are no real solutions.
  3. a) 22 + 14i b) −^75 +^115 i c) 8√3 + i√ 3 d) −^72
  4. a) x = 4, − 2 ± 2 i√ 3 b) x = ± 1 , ±i c) x = − 2 , 1 , 1 ± i√ 3 , − 21 ±

2 i

  1. P (x) = (x − 3)(x + 1)(x − i)(x + i)
  2. see graphs in text sec 3.6 (39, 53,47), chapter 3 review (69), sec 3.6 (43)
  3. It does not exist. Since the complex conjugates are also zeros, the polynomial must be of degree 8.
  4. Let P (x) = x^101 − x^4 + 2. The remainder from dividing P (x) by x + 1 is 0 since P (−1) = (−1)^101 − (−1)^4 + 2 = 0.
  5. (1, 26), (2, 68), (5, 770), (− 2 , −28)
  6. See graph for Chapter 3 review number 31
  7. x = 0, ±√ 2 , ±i
  8. degree 8 as x → −∞, y → −∞ as x → ∞, y → −∞ y−intercept (0, e)
  9. b =^92
  10. a) at most 2 x−intercepts, since the numerator is a quadratic b) at most 3 vertical asymptotes, since the denominator is a cubic

| b

c d