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Zeros of Polynomials - Calculus - Exercise, Exercises of Calculus

Aligarh Muslim UniversityCalculus

These are the notes of Exercise of Calculus. Key important points are: Zeros of Polynomials, Requested Function, Polynomial Function, Leading Coefficient, Degree, Rational Zeros, Real Zeros, Exact Values, Zero as Rational, Irrational

Typology: Exercises

2012/2013

Uploaded on 02/11/2013

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Zeros of Polynomials
Findtherequestedfunction.
1) Findthepolynomialfunctionwithleadingcoefficient2;degree3;and-1
,
2
,
and3aszeros.
A) f(x)=2(x+1)(x-2)(x-3) B) f(x) =(x+1)(x-2)(x-3)
C) f(x)=2(x-1)(x+2)(x+3) D) f(x) =(x-1)(x+2)(x+3)
2) Findthepolynomialfunctionwithleadingcoefficient3;degree3;and-4
,
2
,
and-3aszeros.
A) f(x)=3(x-4)(x+2)(x-3) B) f(x) =(x-4)(x+2)(x-3)
C) f(x)=3(x+4)(x-2)(x+3) D) f(x) =(x+4)(x-2)(x+3)
3) Findthepolynomialfunctionwithleadingcoefficient2;degree4;and7
,
-5
,
2
,
and-2aszeros.
A) f(x)=2(x-7)(x+5)(x-2)(x+2) B) f(x) =2x(x-7)(x+5)(x-2)(x+2)
C) f(x)=2(x+7)(x-5)(x+2)(x-2) D) f(x) =(x-7)(x+5)(x-2)(x+2)
4) Findthepolynomialfunctionwithleadingcoefficient3;degree3;and-3,4,and2
3aszeros.
A) f(x)=3x3-5x2-34x+24 B) f(x)=3x3+5x2-21x-36
C) f(x)=3x3-5x2-21x+36 D) f(x)=3x3+5x2-34x-24
Findallrationalzeros.
5) f(x)=x3+4x2-27x-90
A) -4
,
-6
,
10 B) 4
,
6
,
-10 C) -3
,
-6
,
5D)3
,
6
,
-5
6) f(x)=x3-8x2+4x+48
A) 5
,
7
,
-2B)
-4
,
-6
,
2C)4
,
6
,
-2D)
-5
,
-7
,
2
7) f(x)=4x3-16x2-x+4
A) 1,-1,4B)
1
2,-1
2,4C)2,-2,4D)
1
2,-1
2,-4
8) f(x)=12x3+73x2+5x-6
A) -1
3,1
4,-6B)
-3,-4,-6C)
-3,4,-6D)
-1
12 ,1,-6
9) f(x)=10x3+63x2+17x-6
A) -1
2,1
5,-6B)
-2,5,-6C)
-1
10 ,1,-6D)
-2,-5,-6
10) f x =x4+2x3+2x2-4x-8
A) Norationalzeros B) -2
,
-3C)2
,
-1D)
-2
,
2
pf3
pf4
pf5

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Zeros of Polynomials

Find the requested function.

  1. Find the polynomial function with leading coefficient 2; degree 3; and - 1, 2, and 3 as zeros. A) f(x) = 2(x + 1)(x - 2)(x - 3) B) f(x) = (x + 1)(x - 2)(x - 3) C) f(x) = 2(x - 1)(x + 2)(x + 3) D) f(x) = (x - 1)(x + 2)(x + 3)

  2. Find the polynomial function with leading coefficient 3; degree 3; and - 4, 2, and - 3 as zeros. A) f(x) = 3(x - 4)(x + 2)(x - 3) B) f(x) = (x - 4)(x + 2)(x - 3) C) f(x) = 3(x + 4)(x - 2)(x + 3) D) f(x) = (x + 4)(x - 2)(x + 3)

  3. Find the polynomial function with leading coefficient 2; degree 4; and 7, - 5, 2, and - 2 as zeros. A) f(x) = 2(x - 7)(x + 5)(x - 2)(x + 2) B) f(x) = 2x(x - 7)(x + 5)(x - 2)(x + 2) C) f(x) = 2(x + 7)(x - 5)(x + 2)(x - 2) D) f(x) = (x - 7)(x + 5)(x - 2)(x + 2)

  4. Find the polynomial function with leading coefficient 3 ; degree 3; and - 3, 4, and 2 3

as zeros.

A) f(x) = 3x3- 5x2^ - 34x + 24 B) f(x) = 3x3+ 5x2^ - 21x - 36 C) f(x) = 3x3- 5x2^ - 21x + 36 D) f(x) = 3x3+ 5x2^ - 34x - 24

Find all rational zeros.

  1. f(x) = x3^ + 4x2^ - 27x - 90 A) - 4, - 6, 10 B) 4, 6, - 10 C) - 3, - 6, 5 D) 3, 6, - 5

  2. f(x) = x3^ - 8x2^ + 4x + 48 A) 5, 7, - 2 B) - 4, - 6, 2 C) 4, 6, - 2 D) - 5, - 7, 2

  3. f(x) = 4x3^ - 16x2^ - x + 4 A) 1, - 1, 4 B)^1 2

, 4 C) 2, - 2, 4 D)^1

  1. f(x) = 12x3^ + 73x2^ + 5x - 6 A) - 13 , 14 , - 6 B) - 3, - 4, - 6 C) - 3, 4, - 6 D) - 12 1 , 1, - 6

  2. f(x) = 10x3^ + 63x2^ + 17x - 6 A) - 1 2

, - 6 B) - 2, 5, - 6 C) - 1

, 1, - 6 D) - 2, - 5, - 6

  1. f x = x4^ + 2x3^ + 2x2^ - 4x - 8 A) No rational zeros B) - 2, - 3 C) 2, - 1 D) - 2, 2

PreCalculus

Find all of the real zeros of the function. Give exact values whenever possible. Identify each zero as rational or irrational.

  1. f(x) = x3^ - 2x2^ - 6x + 12 A) 2 (rational), 2 6 (irrational), and - 2 6 (irrational) B) - 2 (rational), 6 (irrational), and - 6 (irrational) C) 2 (rational), 6 (irrational), and - 6 (irrational) D) 2 (rational), 6 (rational), and - 6 (rational)

  2. f(x) = x3^ - 7x2^ + 4x + 30 A) 5 (rational), 2 + 7 (irrational), and 2 - 7 (irrational) B) 5 (rational), 7 (irrational), and - 7 (irrational) C) 5 (rational), 8 (rational), - 6 (rational) D) 5 (rational), 1 + 7 (irrational), and 1 - 7 (irrational)

  3. f(x) = x3^ - 7x2^ + 10x + 6 A) 3 (rational), 2 + 6 (irrational), and 2 - 6 (irrational) B) 3 (rational), 6 (irrational), and - 6 (irrational) C) 3 (rational), 1 + 6 (irrational), and 1 - 6 (irrational) D) 3 (rational), - 2 + 6 (irrational), and - 2 - 6 (irrational)

Write the polynomial in standard form and identify the zeros of the function.

  1. f(x) = (x - 5i)(x +5i) A) f(x) = x2^ - 25; zeros ± 5i B) f(x) = x2^ + 5ix + 25; zeros ± 5 C) f(x) = x2^ + 25; zeros ± 5 D) f(x) = x2^ + 25; zeros ± 5i

  2. f(x) = (x - 1)(x - 7i)(x + 7i) A) f(x) = x3^ - x2^ + 49x - 49; zeros 1, ± 7i B) f(x) = x3^ - x2^ + 7x - 7; zeros 1, ± 7i C) f(x) = x3^ - x2^ - 7x + 7; zeros 1, ± 7i D) f(x) = x3^ - x2^ - 7x + 7; zeros - 1, ± 7i

  3. f(x) = (x + 2)(x + 2)(x + 3i)(x - 3i) A) f(x) = x4^ + 4x3^ + 13x2^ + 36x + 36; zeros 2 (mult. 2), ± 3i B) f(x) = x4^ + 4x3^ - 5x2^ - 36x - 36; zeros - 2 (mult. 2), ± 3 C) f(x) = x4^ + 4x3^ + 13x2^ + 36x + 36; zeros - 2 (mult. 2), ± 3i D) f(x) = x4^ + 13x2^ + 36; zeros 2 (mult. 2), ± 3

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

  1. 4i and 3 A) f(x) = x4^ - 13x2^ - 48 B) f(x) = x4^ - 26x2^ + 48 C) f(x) = x4^ + 26x2^ + 48 D) f(x) = x4^ + 13x2^ - 48

  2. 7, - 11, and 2 + 8i A) f(x) = x4^ - 145x2^ + 580x - 5236 B) f(x) = x4^ - 25x2^ + 580x - 5236 C) f(x) = x4^ - 9x3^ - 56x2^ + 290x - 5236 D) f(x) = x4^ - 9x3^ + 56x2^ - 290x + 5236

PreCalculus

    • 2i is a zero of f(x) = x4^ - 45x2^ - 196 A) 2i, 7i, - 7i B) 2i, 14i, - 14i C) 2i, 14, - 14 D) 2i, 7, - 7

Answer Key

Testname: 1_ZEROS_OF_POLYNOMIALS

1) A

2) C

3) A

4) A

5) C

6) C

7) B

8) A

9) A

10) A

11) C

12) D

13) A

14) D

15) B

16) C

17) D

18) B

19) A

20) B

21) B

22) B

23) D

24) C

25) D

26) C

27) C

28) B

29) D