Review Test Polynomials

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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Determine whether the function is a polynomial function.

1) f(x) = 7 -1x3 1)

A) Yes B) No

2) f(x) =4

x5 - x2 + 3 2)A) Yes B) No

3) f(x) = x4 + 3x3 + 7 3)A) No B) Yes

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis andturns around, at each intercept.

4) f(x) = x4 - 49x2 4)A) 0, touches the x-axis and turns around;

7, crosses the x-axis;-7, crosses the x-axis

B) 0, touches the x-axis and turns around;49, touches the x-axis and turns around

C) 0, crosses the x-axis;7, crosses the x-axis;-7, crosses the x-axis

D) 0, touches the x-axis and turns around;49, crosses the x-axis

5) x5 - 19x3 + 48x = 0 5)A) 0, touches the x-axis and turns around;

4, crosses the x-axis;-4, crosses the x-axis;

3, crosses the x-axis;- 3, crosses the x-axis

B) 0, touches the x-axis and turns around;16, touches the x-axis and turns around;3, touches the x-axis and turns around

C) 0, crosses the x-axis;4, crosses the x-axis;-4, crosses the x-axis;

3, crosses the x-axis;- 3, crosses the x-axis

D) 0, crosses the x-axis;16, touches the x-axis and turns around;3, touches the x-axis and turns around

6) f(x) = -x2(x + 9)(x2 - 1) 6)A) 0, touches the x-axis and turns around;

-9, crosses the x-axis;-1, crosses the x-axis;1, crosses the x-axis

B) 0, touches the x-axis and turns around;9, crosses the x-axis;-1, touches the x-axis and turns around;1, touches the x-axis and turns around

C) 0, touches the x-axis and turns around;-9, crosses the x-axis;1, touches the x-axis and turns around

D) 0, crosses the x-axis;-9, crosses the x-axis;-1, crosses the x-axis;1, crosses the x-axis

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7) f(x) = -x3(x + 3)2(x - 9) 7)A) 0, crosses the x-axis;

3, touches the x-axis and turns around;-9, crosses the x-axis

B) 0, crosses the x-axis;-3, touches the x-axis and turns around;9, crosses the x-axis

C) 0, touches the x-axis and turns around;3, crosses the x-axis;9, crosses the x-axis

D) 0, touches the x-axis and turns around;-3, touches the x-axis and turns around;9, crosses the x-axis

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end behaviorto match the function with its graph.

8) f(x) = -4x2 - 2x + 1 8)A) rises to the left and rises to the right B) falls to the left and falls to the right

C) rises to the left and falls to the right D) falls to the left and rises to the right

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9) f(x) = -6x3 - 2x2 + 3x + 2 9)A) rises to the left and falls to the right B) falls to the left and falls to the right

C) rises to the left and rises to the right D) falls to the left and rises to the right

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10) f(x) = 2x4 - 4x2 10)A) rises to the left and falls to the right B) rises to the left and rises to the right

C) falls to the left and falls to the right D) falls to the left and rises to the right

Use the Leading Coefficient Test to determine the end behavior of the polynomial function.11) f(x) = -4x4 + 3x3 + 4x2 + 3x + 5 11)

A) rises to the left and falls to the right B) falls to the left and rises to the rightC) rises to the left and rises to the right D) falls to the left and falls to the right

12) f(x) = -3x3 - 2x2 + 3x - 3 12)A) rises to the left and falls to the right B) rises to the left and rises to the rightC) falls to the left and rises to the right D) falls to the left and falls to the right

13) f(x) = (x + 2)(x + 3)(x + 5)3 13)A) falls to the left and rises to the right B) rises to the left and falls to the rightC) rises to the left and rises to the right D) falls to the left and falls to the right

14) f(x) = -x2(x - 1)(x + 2) 14)A) falls to the left and falls to the right B) falls to the left and rises to the rightC) rises to the left and rises to the right D) rises to the left and falls to the right

Find the zeros of the polynomial function.15) f(x) = x3 + 2x2 - x - 2 15)

A) x = - 2, x = 2 B) x = 4C) x = -1, x = 1, x = - 2 D) x = 1, x = - 2, x = 2

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16) f(x) = x3 + 5x2 - 4x - 20 16)A) x = 5, x = -2, x = 2 B) x = -5, x = -2, x = 2C) x = -2, x = 2 D) x = -5, x = 4

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses thex-axis or touches the x-axis and turns around, at each zero.

17) f(x) = 3(x + 1)(x + 2)3 17)A) -1, multiplicity 1, crosses x-axis; -2, multiplicity 3, touches x-axis and turns aroundB) 1, multiplicity 1, crosses x-axis; 2, multiplicity 3, crosses x-axisC) -1, multiplicity 1, crosses x-axis; -2, multiplicity 3, crosses x-axisD) 1, multiplicity 1, touches x-axis; 2, multiplicity 3, touches x-axis and turns around

18) f(x) = 4(x2 + 2)(x + 6)2 18)A) -2, multiplicity 1, crosses the x-axis; -6, multiplicity 2, crosses the x-axisB) -2, multiplicity 1, crosses the x-axis; -6, multiplicity 2, touches the x-axis and turns around.C) -6, multiplicity 2, touches the x-axis and turns aroundD) -6, multiplicity 2, crosses the x-axis

19) f(x) =15

x2(x2 - 5)(x - 5) 19)

A) 0, multiplicity 2, touches x-axis and turns around;5, multiplicity 1, crosses x-axis5, multiplicity 2, touches x-axis and turns around

B) 0, multiplicity 2, touches x-axis and turns around;5, multiplicity 1, crosses x-axis

C) 0, multiplicity 2, crosses x-axis;5, multiplicity 1, touches x-axis and turns around;

5, multiplicity 1, touches x-axis and turns around;- 5, multiplicity 1, touches x-axis and turns around

D) 0, multiplicity 2, touches x-axis and turns around;5, multiplicity 1, crosses x-axis;

5, multiplicity 1, crosses x-axis;- 5, multiplicity 1, crosses x-axis

20) f(x) = x3 + x2 - 42x 20)A) 0, multiplicity 1, crosses the x-axis

7, multiplicity 1, crosses the x-axis-6, multiplicity 1, crosses the x-axis

B) 0, multiplicity 1, crosses the x-axis- 7, multiplicity 1, crosses the x-axis6, multiplicity 1, crosses the x-axis

C) - 7, multiplicity 2, touches the x-axis and turns around6, multiplicity 1, crosses the x-axis

D) 0, multiplicity 1, touches the x-axis and turns around;- 7, multiplicity 1, touches the x-axis and turns around;6, multiplicity 1, touches the x-axis and turns around

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Write the equation of a polynomial function with the given characteristics. Use a leading coefficient of 1 or -1 and makethe degree of the function as small as possible.

21) Crosses the x-axis at -4, 0, and 2; lies below the x-axis between -4 and 0; lies above the x-axisbetween 0 and 2.

21)

A) f(x) = x3 + 2x2 - 8x B) f(x) = x3 - 2x2 - 8xC) f(x) = -x3 - 2x2 + 8x D) f(x) = - x3 + 2x2 + 8x

22) Touches the x-axis at 0 and crosses the x-axis at 3; lies above the x-axis between 0 and 3. 22)A) f(x) = -x3 - 3x2 B) f(x) = x3 + 3x2

C) f(x) = x3 - 3x2 D) f(x) = -x3 + 3x2

Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the givenintegers.

23) f(x) = 7x5 - 2x3 + 5x2 + 9; between -2 and -1 23)A) f(-2) = 179 and f(-1) = 9; no B) f(-2) = -179 and f(-1) = -9; noC) f(-2) = -179 and f(-1) = 9; yes D) f(-2) = 179 and f(-1) = -9; yes

24) f(x) = 2x4 + 6x3- 4x - 1; between -3 and -2 24)A) f(-3) = -11 and f(-2) = -9; no B) f(-3) = 11 and f(-2) = -9; yesC) f(-3) = -11 and f(-2) = 9; yes D) f(-3) = 11 and f(-2) = 9; no

Determine the maximum possible number of turning points for the graph of the function.25) f(x) = -x2 - 6x - 7 25)

A) 2 B) 3 C) 0 D) 1

26) f(x) = 8x3 - 8x2 - 5x - 22 26)A) 0 B) 8 C) 2 D) 3

27) f(x) = x6 + 8x7 27)A) 1 B) 6 C) 7 D) 8

28) f(x) = (x + 2)(x - 1)(7x + 2) 28)A) 3 B) 0 C) 7 D) 2

29) f(x) = (5x - 5)2( x2 - 1)(x + 1) 29)A) 2 B) 5 C) 25 D) 4

Graph the polynomial function.

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30) f(x) = x4 - 4x2 30)

A) B)

C) D)

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31) f(x) = x3 + 5x2 - x - 5 31)

A) B)

C) D)

Divide using long division.

32) 6m3 + 5m2 - 9m + 10m + 2

32)

A) m2 + 7m + 6 B) 6m2 - 7m + 5 C) m2 + 8m + 9 D) 6m2 + 7m + 5

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33) (6x3 - 2) ÷ (3x - 1) 33)

A) 2x2 +23

x +29

+16

9(3x - 1) B) 2x2 +23

x +29

-16

9(3x - 1)

C) 2x2 +23

x +29 D) 2x2 -

23

x +29

34) 8y4 + 12y3 - 2y2y2 + y

34)

A) 4y2 + 4y - 2 B) 4y2 + 4y -6y

2y2 + y

C) 4y2 + 6y -2y

2y2 + yD) 4y2 + 8y + 4 +

2y2y2 + y

Divide using synthetic division.

35) -4x3 - 28x2 - 21x + 18x + 6

35)

A) 4x2 - 6x + 3 B) -23

x2 -143

x -72

C) 4x2 + 6x - 3 D) -4x2 - 4x + 3

36) x5 + x3 + 1x + 3

36)

A) x4 - 2x2 +7

x + 3 B) x4 - 2 +7

x + 3

C) x4 - 3x3 + 9x2 - 26x + 78 +-233x + 3 D) x4 - 3x3 + 10x2 - 30x + 90 +

-269x + 3

37) (x5 - 4x4 - 9x3 + x2 - x + 21) ÷ (x + 2) 37)

A) x4 - 6x3 + 3x2 - 5x + 9 +3

x + 2 B) x4 - 6x3 + 3x2 - 6x - 10 +6

x + 2

C) x4 - 6x3 + 3x2 - 5x - 9 +3

x + 2 D) x4 - 6x3 + 3x2 - 6x + 9 +6

x + 2

Use synthetic division and the Remainder Theorem to find the indicated function value.38) f(x) = 2x3 - 6x2 - 3x + 15; f(-2) 38)

A) -13 B) -19 C) -31 D) -10

39) f(x) = x5 + 8x4 + 2x3 + 4; f(-2) 39)A) 116 B) -84 C) 84 D) 24

40) f(x) = x4 + 5x3 - 6x2 - 2x - 8; f 12 40)

A) 15716 B) -

15732 C) -

15716 D) -

798

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Solve the problem.41) Use synthetic division to divide f(x) = x3 + 15x2 + 71x + 105 by x + 7. Use the result to find all zeros

of f.41)

A) {-7 , 3, 5} B) {7, -3, -5} C) {7, 3, 5} D) {-7, -3, -5}

42) Solve the equation 12x3 - 65x2 + 24x + 10 = 0 given that 23

is a root. 42)

A) 23

, -14

, 5 B) 23

, 14

, -5 C) 23

, -54

, 1 D) 23

, 54

, -1

Use the Rational Zero Theorem to list all possible rational zeros for the given function.43) f(x) = x5 - 4x2 + 6x + 5 43)

A) ± 1, ± 5 B) ± 5, ± 15 C) ± 1, ± 1

5 D) ± 14

, ± 54

, ± 5

44) f(x) = x4 + 7x3 - 5x2 + 2x - 12 44)A) ± 1, ± 2, ± 3, ± 4, ± 6, ± 12

B) ± 112

, ± 1, ± 12

C) ± 1, ± 12

, ± 13

, ± 14

, ± 16

, ± 112

D) ± 12

, ± 13

, ± 14

, ± 16

, ± 112

, ± 1, ± 2, ± 3, ± 4, ± 6, ± 12

45) f(x) = 7x3 - x2 + 3 45)

A) ± 17

, ± 37

, ± 1, ± 3 B) ± 17

, ± 13

, ± 1, ± 3, ± 7

C) ± 17

, ± 37

, ± 1, ± 3, ± 7 D) ± 13

, ± 73

, ± 1, ± 7

46) f(x) = 6x4 + 2x3 - 4x2 + 2 46)

A) ± 16

, ± 13

, ± 12

, ± 1, ± 2 B) ± 16

, ± 13

, ± 12

, ± 23

, ± 1, ± 2

C) ± 16

, ± 13

, ± 12

, ± 23

, ± 1, ± 2, ± 3 D) ± 12

, ± 32

, ± 1, ± 2, ± 3, ± 6

47) f(x) = 6x4 + 3x3 - 3x2 + 3x - 5 47)

A) ± 1, ± 2, ± 3, ± 6, ± 12

, ± 52

, ± 13

, ± 53

, ± 16

, ± 56

B) ± 1, ± 5, ± 15

, ± 25

, ± 35

, ± 65

C) ± 1, ± 5, ± 12

, ± 52

, ± 13

, ± 53

, ± 16

, ± 56

D) ± 1, ± 2, ± 3, ± 6, ± 15

, ± 25

, ± 35

, ± 65

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48) f(x) = 3x4 + 7x3 - 3x2 + 5x - 12 48)

A) ± 1, ± 3, ± 12

, ± 32

, ± 13

, ± 14

, ± 34

, ± 16

, ± 112

B) ±1, ± 2, ± 3, ± 4, ± 6, ± 12, ± 12

, ± 32

, ± 13

, ± 14

, ± 34

, ± 16

, ± 112

C) ±1, ± 2, ± 3, ± 4, ± 6, ± 12, ± 13

, ± 23

, ± 43

D) ±1, ± 2, ± 3, ± 6, ± 12, ± 13

, ± 23

, ± 34

Find a rational zero of the polynomial function and use it to find all the zeros of the function.49) f(x) = 3x3 - x2 - 9x + 3 49)

A) {-3, 3, - 3} B) {3, 3, - 3} C) {13

, 3, - 3} D) {- 13

, 3, - 3}

50) f(x) = x3 + 2x2 - 5x - 6 50)A) {-1} B) {-3} C) {-3, -1, 2} D) {-2, 1, 3}

51) f(x) = x4 + 3x3 - 5x2 - 9x - 2 51)A) {1, -2, -2 + 3, -2 - 3} B) {-1, 3, -2 + 5, -2 - 5}C) {-1, 2, -2 + 3, -2 - 3} D) {-1, -2, -2 + 5, -2 - 5}

52) f(x) = 2x4 - 17x3 + 59x2 - 83x + 39 52)

A) {1, -32

, 2 + 3i, 2 - 3i} B) {1, 32

, 3 + 2i, 3 -2i}

C) {-1, 32

, 3 + 2i, 3 - 2i} D) {-1, -32

, 2 + 3i, 2 - 3i}

Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots.53) 4x3 - 23x2 + 26x + 8 = 0 53)

A) 14

, 2, -4 B) 1, -1, 2 C) -14

, 2, 4 D) - 1, -1, -2

54) 2x3 - x2 - 12x + 6 = 0 54)

A) {2, 6, - 6} B) {-2, 6, - 6} C) {- 12

, 6, - 6} D) {12

, 6, - 6}

55) 2x4 - 19x3 + 74x2 - 127x + 78 = 0 55)

A) {2, -32

, 2 + 3i, 2 - 3i} B) {-2, 32

, 3 + 2i, 3 - 2i}

C) {2, 32

, 3 + 2i, 3 -2i} D) {-2, -32

, 2 + 3i, 2 - 3i}

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56) 3x4 + 16x3 + 56x2 + 56x + 13 = 0 56)

A) {1, +13

, -3 + 2i, -3 - 2i} B) {-1, +13

, -3 + 2i, -3 - 2i}

C) {1, -13

, -2 + 3i, -2 - 3i} D) {-1, -13

, -2 + 3i, -2 - 3i}

Find an nth degree polynomial function with real coefficients satisfying the given conditions.57) n = 3; - 4 and i are zeros; f(-3) = 60 57)

A) f(x) = 6x3 + 24x2 + 6x + 24 B) f(x) = -6x3 - 24x2 + 6x + 24C) f(x) = -6x3 - 24x2 - 6x - 24 D) f(x) = 6x3 + 24x2 - 6x - 24

58) n = 4; 3, 13

, and 1 + 2i are zeros; f(1) = 48 58)

A) f(x) = 3x4 - 16x3 + 38x2 + 168x - 45 B) f(x) = -3x4 + 48x3 - 114x2 + 168x - 45C) f(x) = -6x4 + 48x3 - 114x2 + 168x - 45 D) f(x) = -6x4 + 32x3 - 76x2 + 112x - 30

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the given function.59) f(x) = -5x7 + x3 - x2 + 9 59)

A) 2 or 0 positive zeros, 2 or 0 negative zeros B) 2 or 0 positive zeros, 3 or 1 negative zerosC) 3 or 1 positive zeros, 3 or 1 negative zeros D) 3 or 1 positive zeros, 2 or 0 negative zeros

60) f(x) = 6x5 - 4x2 + x + 4 60)A) 3 or 1 positive zeros, 3 or 1 negative zeros B) 2 or 0 positive zeros, 1 or 0 negative zerosC) 2 or 0 positive zeros, 2 or 0 negative zeros D) 2 or 0 positive zeros, 1 negative zero

61) f(x) = x5 - 1.5x4 - 13.76x3 + 3x2 + 34.42x - 15.397 61)A) 3 or 1 positive zeros, 2 or 0 negative zeros B) 2 or 0 positive zeros, 3 or 1 negative zerosC) 3 or 1 positive zeros, 3 or 1 negative zeros D) 2 or 0 positive zeros, 2 or 0 negative zeros

62) f(x) = 6x8 - 9x7 + x6 - 3x + 18 62)A) 4, 2 or 0 positive zeros, no negative zeros B) 4 or 2 positive zeros, no negative zerosC) 4 positive zeros, no negative zeros D) 4, 2 or 0 positive zeros, 1 negative zeros

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