Unit 1
Review
MHF 4U1 – Advanced Functions
Polynomial Functions Unit Review
Part 2: Solving Polynomial Equations
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Work on the following review questions independently to prepare for your Unit 1 Test.
Don’t forget to review ALL examples from the notes and ALL homework questions while you study!!!
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Polynomial Division (1.9)
1. Divide.
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a)
( x3  x 2  5 x  2)  ( x  2)
b)
(2 x3  7 x 2  7 x  5)  (2 x  1)
c)
(15x  4 x3  9 x 2  3x 4  4)  (3x  4)
The Remainder Theorem (1.10)
2. Use the remainder theorem to determine the remainder for each division.
a)
( x3  4 x 2  2 x  5)  ( x  5)
b)
(4 y 3  y 2  12 y  5)  (4 y  1)
The Factor Theorem (1.11)
3. Factor completely.
a) x  2 x  21x  18
3
2
b) x  5 x  3 x  4
3
2
c) 3 x  10 x  9 x  4
3
2
d) 3 x  13 x  16
3
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2
Solving Polynomial Equations (1.12)
4. Solve for x. Check your answer.
a) x  3 x  2  0
3
b) 3x  2  8 x  7 x
3
3
c) x  13x  36  0
4
2
d) 4 x  2 x  16 x  8 x  0
5. a) Find the family of cubic functions whose x-intercepts are –3, 0, and 2.
b) Find the actual equation of the function if f(-1) = 12.
4
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3
2
Inequalities (1.13-1.14)
6. Solve. Write your solution in interval notation and graph it on a number line.
a)  (3  x)  2(3 x  2)
11  3x  4  7
2
c) x  6 x  7  0
4
2
d)
3x  2
b)
7. Solve by graphing.
a) ( x  3)( x  1)  0
b) x  5 x  x  5
3
2
8. Determine values of x for which the graph of f ( x)  x  5 x
HINT: How can we represent “below” using math language?
3
2
 2 x is below that of f ( x)  8 .
Polynomial Functions Unit Review
Part 2: Solving Polynomial Equations
ANSWERS
1. a) ( x  2)( x 2  3x  1)
2. a) –10
b) –8
3. a) ( x  1)( x  6)( x  3)
d) ( x  1)(3x  4)( x  4)
4. a) –1, 2
b) 1,
b) (2 x  1)( x 2  3x  5)
c) (3x  4)( x3  3x  1)
b) ( x  4)( x 2  x  1)
c) ( x  1)(3x  1)( x  4)
 5  65  5  65
,
or 1, -0.31, 1.31 c) 2, –2, –3, 3
 10
 10
d) 0, 2, ½ , –2
5. a) f ( x)  kx( x  3)( x  2) b) f ( x)  2 x( x  3)( x  2)
6. a) ( 1, )
b) (5,1]
c) (,7]  [1, )
4
3
d) [ ,  )
7. a) [3,1]
b) (,1)  (1, 5)
8. (,4)  (2,1)
Polynomial Functions Unit Review
Part 2: Solving Polynomial Equations
ANSWERS
1. a) ( x  2)( x 2  3x  1)
2. a) –10
b) –8
3. a) ( x  1)( x  6)( x  3)
d) ( x  1)(3x  4)( x  4)
4. a) –1, 2
b) 1,
b) (2 x  1)( x 2  3x  5)
c) (3x  4)( x3  3x  1)
b) ( x  4)( x 2  x  1)
c) ( x  1)(3x  1)( x  4)
 5  65  5  65
,
or 1, -0.31, 1.31 c) 2, –2, –3, 3
 10
 10
d) 0, 2, ½ , –2
5. a) f ( x)  kx( x  3)( x  2) b) f ( x)  2 x( x  3)( x  2)
6. a) ( 1, )
b) (5,1]
c) (,7]  [1, )
7. a) [3,1]
b) (,1)  (1, 5)
8. (,4)  (2,1)
4
3
d) [ ,  )