MPJNS-MHF4U1-ASSIGNMENT CHAPTER 2 A



MHF4U1-ASSIGNMENT CHAPTER 2 A NAME:___________________________

True/False

Indicate whether the statement is true or false.

____ 1. When performing long division of a polynomial by a linear binomial, the degree of the remainder is always smaller than the degree of the divisor.

____ 2. If P(–3) = 0 for a polynomial P(x), then x + 3 is a factor of P(x).

____ 3. For a polynomial equation P(x) = 0, if P(x) is not factorable, then P(x) = 0 has no real roots.

____ 4. All quartic polynomial equations have at least one real solution.

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 5. If x3 – 4x2 + 5x – 6 is divided by x – 1, then the restriction on x is

|a. |x [pic] –4 |c. |x [pic] 1 |

|b. |x [pic] –1 |d. |no restrictions |

____ 6. What is the remainder when x4 + 2x2 – 3x + 7 is divided by x + 2?

|a. |25 |c. |37 |

|b. |13 |d. |9 |

____ 7. If 6x4 – 2x3 – 21x2 + 7x + 8 is divided by 3x – 1 to give a quotient of 2x3 – 7x and

a remainder of 8, then which of the following is true?

|a. |[pic] |

|b. |6x4 – 2x3 – 21x2 + 7x + 8 = (3x – 1)(2x3 – 7x) + 8 |

|c. |[pic] |

|d. |all of the above |

____ 8. When P(x) = 4x3 – 4x + 1 is divided by 2x – 3, the remainder is

|a. |[pic] |c. |[pic] |

|b. |P(3) = 97 |d. |[pic] |

____ 9. For a polynomial P(x), if P[pic] = 0, then which of the following must be a factor of P(x)?

|a. |[pic] |c. |5x + 3 |

|b. |3x + 5 |d. |5x – 3 |

____ 10. Which of the following binomials is a factor of x3 – 6x2 + 11x – 6?

|a. |x – 1 |c. |x + 7 |

|b. |x + 1 |d. |2x + 3 |

____ 11. Which set of values for x should be tested to determine the possible zeros of x3 – 2x2 + 3x – 12?

|a. |1, 2, 3, 4, 6, and 12 |c. |±1, ±2, ±3, ±4, and ±6 |

|b. |±1, ±2, ±3, ±4, ±6, and ±12 |d. |±2, ±3, ±4, ±6, and ±12 |

____ 12. Determine the value of k so that x – 3 is a factor of x3 – 3x2 + x + k.

|a. |k = 3 |c. |k = 1 |

|b. |k = –3 |d. |k = –1 |

____ 13. Find k if 2x + 1 is a factor of kx3 + 7x2 + kx – 3.

|a. |k = –2 |c. |k = [pic] |

|b. |k = 2 |d. |none of the above |

____ 14. Which of the following is the fully factored form of x3 + 3x2 – x – 3?

|a. |(x + 3)(x2 – 1) |c. |x2(x + 3) – (x + 3) |

|b. |(x – 1)(x + 1)(x + 3) |d. |(x2 – 1)(x – 3) |

____ 15. Which of the following is the fully factored form of x3 – 6x2 – 6x – 7?

|a. |(x – 7)(x + 1)2 |c. |(x – 7)(x2 + x + 1) |

|b. |(x – 7)(x + 1)(x – 1) |d. |(x – 6)(x + 1)(x – 1) |

____ 16. Which of the following is the factored form of x4 – 2x2 – 3?

|a. |(x – 1)(x + 1)(x – 3) |c. |(x2 + 1)(x2 – 3) |

|b. |(x2 – 1)(x + 3) |d. |none of the above |

____ 17. One root of the equation x3 + 2x – 3x2 – 6 = 0 is

|a. |–3 |c. |3 |

|b. |–1 |d. |1 |

____ 18. What is the maximum number of real distinct roots that a quartic equation can have?

|a. |infinitely many |c. |2 |

|b. |4 |d. |none of the above |

____ 19. If 2 is one root of the equation 4x3 + kx – 24 = 0, then the value of k is

|a. |–1 |c. |8 |

|b. |–4 |d. |impossible to determine |

____ 20. Based on the graph of f(x) = x4 – 2x3 + 3x + 2 shown, what are the real roots of x4 – 2x3 + 3x + 2 = 0?

[pic]

|a. |2 |c. |impossible to determine |

|b. |–2, –1, 1, 2 |d. |no real roots |

____ 21. Which of the following graphs of polynomial functions corresponds to a cubic polynomial equation with roots –2, 3, and 4?

|a. | |c. | |

| |[pic] | |[pic] |

|b. | |d. | |

| |[pic] | |[pic] |

Completion

Complete each statement.

22. If P(x) is divided by ax – b, then the _______________ is P[pic].

23. If x(4x – 3)(x + 1) = 0, then the solutions for x are _______________.

24. The x-intercepts of the graph of a polynomial function correspond to the _______________ of the related polynomial equation.

Matching

Match the correct term with the correct part of the statement.

[pic]

|a. |quotient |c. |divisor |

|b. |remainder |d. |dividend |

____ 25. 4x3 + x – 3

____ 26. 2x – 1

____ 27. 2x2 + x + 1

____ 28. –2

Short Answer

29. a) Use long division to divide x3 + 3x2 – 7 by x + 2. Express the result in quotient form.

b) Identify any restrictions on the variable.

c) Write the corresponding statement that can be used to check the division.

d) Verify your answer.

30. Factor fully.

a) 29x2 – 21 + 10x4

b) 2(x + 1)2 – 32

c) 6x3 – 7x2 – 12x + 13

31. Factor fully.

a) 2(n – 1)2 – 4(n – 1) + 2

b) 2x4 + 7x3 – 10x2 – 32x

c) x3 – x2 – x + 1

32. Factor fully.

a) x2(x – 2)(x + 2) + 3x + 6

b) 16x4 – (x + 1)2

c) 2x3 + 5x2 – 14x – 8

33. Solve.

a) 3x3 + 2x2 – 8x + 3 = 0

b) 2x3 + x2 – 10x – 5 = 0

c) 5x4 = 7x2 – 2

34. Solve by factoring.

a) 8x3 – 36x2 + 46x – 15 = 0

b) x4 + 3x2 – 28 = 0

c) 2x4 – 54x = 0

Problem

35. The polynomial 6x3 + mx2 + nx – 5 has a factor of x + 1. When divided by x – 1, the remainder is –4. What are the values of m and n?

36. Factor 2x4 – 7x3 – 41x2 – 53x – 21 fully.

37. Show that x + a is a factor of the polynomial P(x) = (x + a)4 + (x + c)4 – (a – c)4.

38. Given that –2 is a root of x3 + x = –4x2 + 6, find the other root(s).

39. The height of a square-based box is 4 cm more than the side length of its square base. If the volume of the box is 225 cm3, what are its dimensions?

40. Amit has designed a rectangular storage unit to hold large factory equipment. His scale model has dimensions 1 m by 2 m by 4 m. By what amount should he increase each dimension to produce an actual storage unit that is 9 times the volume of his scale model?

MPJNS-MHF4U1-ASSIGNMENT CHAPTER 2 A

Answer Section

TRUE/FALSE

1. ANS: T PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions

KEY: remainder

2. ANS: T PTS: 1 DIF: 1 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C3.1 TOP: Polynomial and Rational Functions

KEY: factor theorem

3. ANS: F PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.3 LOC: C3.3 TOP: Polynomial and Rational Functions

KEY: polynomial equation, real roots

4. ANS: F PTS: 1 DIF: 1

REF: Knowledge and Understanding; Thinking OBJ: Section 2.3

LOC: C3.3 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots

MULTIPLE CHOICE

5. ANS: C PTS: 1 DIF: 1 REF: Knowledge and Understanding

OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions

KEY: restriction

6. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions

KEY: remainder theorem

7. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions

KEY: quotient, remainder

8. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions

KEY: remainder theorem

9. ANS: C PTS: 1 DIF: 1 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C3.1 TOP: Polynomial and Rational Functions

KEY: factor theorem

10. ANS: A PTS: 1 DIF: 1

REF: Knowledge and Understanding; Application OBJ: Section 2.2

LOC: C3.2 TOP: Polynomial and Rational Functions

KEY: factor theorem, integral zero theorem

11. ANS: B PTS: 1 DIF: 1 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions

KEY: integral zero theorem

12. ANS: B PTS: 1 DIF: 2

REF: Knowledge and Understanding; Application OBJ: Section 2.2

LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factor theorem

13. ANS: A PTS: 1 DIF: 3

REF: Knowledge and Understanding; Application OBJ: Section 2.2

LOC: C3.2 TOP: Polynomial and Rational Functions KEY: factor theorem

14. ANS: B PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions

KEY: factored form

15. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions

KEY: factored form

16. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions

KEY: factored form

17. ANS: C PTS: 1 DIF: 3 REF: Knowledge and Understanding

OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions

KEY: polynomial equation, real roots

18. ANS: B PTS: 1 DIF: 1

REF: Knowledge and Understanding; Thinking OBJ: Section 2.3

LOC: C3.3 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots

19. ANS: B PTS: 1 DIF: 2

REF: Knowledge and Understanding; Application OBJ: Section 2.3

LOC: C3.4 TOP: Polynomial and Rational Functions KEY: polynomial equation, real roots

20. ANS: D PTS: 1 DIF: 1 REF: Knowledge and Understanding

OBJ: Section 2.3 LOC: C3.3 TOP: Polynomial and Rational Functions

KEY: polynomial equation, real roots

21. ANS: A PTS: 1 DIF: 1 REF: Knowledge and Understanding

OBJ: Section 2.3 LOC: C3.3 TOP: Polynomial and Rational Functions

KEY: polynomial equation, real roots

COMPLETION

22. ANS: remainder

PTS: 1 DIF: 1 REF: Knowledge and Understanding

OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions

KEY: remainder theorem

23. ANS: [pic]

PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions

KEY: polynomial equation

24. ANS:

roots

solutions

PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.3 LOC: C3.3 TOP: Polynomial and Rational Functions

KEY: x-intercepts, polynomial equation

MATCHING

25. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions

KEY: quotient, remainder, divisor, dividend

26. ANS: C PTS: 1

27. ANS: A PTS: 1

28. ANS: B PTS: 1

SHORT ANSWER

29. ANS:

a) [pic]

b) x [pic] –2

c) x3 + 3x2 – 7 = (x + 2)(x2 + x – 2) – 3

d) Expand to verify.

PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.1 LOC: C3.1 TOP: Polynomial and Rational Functions

KEY: long division, restriction

30. ANS:

a) (2x2 + 7)(5x2 – 3)

b) 2(x – 3)(x + 5)

c) (x – 1)(6x2 – x – 13)

PTS: 1 DIF: 3 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions

KEY: factor polynomial expression NOT: A variety of factoring techniques is required.

31. ANS:

a) 2(n – 2)2

b) x(x + 2)(2x2 + 3x – 16)

c) (x + 1)(x – 1)2

PTS: 1 DIF: 3 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions

KEY: factor polynomial expression NOT: A variety of factoring techniques is required.

32. ANS:

a) (x + 2)(x + 1)(x2 – 3x + 3)

b) (4x2 – x – 1)(4x2 + x + 1)

c) (2x + 1)(x – 2)(x + 4)

PTS: 1 DIF: 3 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions

KEY: factor polynomial expression NOT: A variety of factoring techniques is required.

33. ANS:

a) [pic]

b) [pic]

c) [pic]

PTS: 1 DIF: 3 REF: Knowledge and Understanding

OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions

KEY: polynomial equation NOT: A variety of factoring techniques is required.

34. ANS:

a) [pic]

b) –2, 2

c) 0, 3

PTS: 1 DIF: 3 REF: Knowledge and Understanding

OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions

KEY: polynomial equation NOT: A variety of factoring techniques is required.

PROBLEM

35. ANS:

Let P(x) = 6x3 +mx2 + nx – 5.

By the factor theorem, since x + 1 is a factor of P(x), then P(–1) = 0.

[pic]

By the remainder theorem, when P(x) is divided by x – 1, the remainder is P(1).

Solve P(1) = –4.

[pic]

Solve the system of equations.

[pic]

Substitute m = 3 into equation (2), to find that n = –8.

PTS: 1 DIF: 4 REF: Knowledge and Understanding; Application; Thinking

OBJ: Sections 2.1, 2.2 LOC: C3.1, C3.2 TOP: Polynomial and Rational Functions

KEY: factor theorem, remainder theorem

36. ANS:

Let P(x) = 2x4 – 7x3 – 41x2 – 53x – 21.

By the rational zero theorem, possible values of [pic] are ±1, ±3, ±7, ±21, [pic], and [pic]. Test the values to find a zero.

Since x = –1 is a zero of P(x), x + 1 is a factor. Divide to determine the other factor.

2x4 – 7x3 – 41x2 – 53x – 21 = (x + 1)(2x3 – 9x2 – 32x – 21)

Factor 2x3 – 9x2 – 32x – 21 using a similar method.

2x3 – 9x2 – 32x – 21 = (x + 1)(2x2 – 11x – 21)

= (x + 1)(2x + 3)(x – 7)

So, P(x) = 2x4 – 7x3 – 41x2 – 53x – 21 = (x + 1)2(2x + 3)(x – 7).

PTS: 1 DIF: 3 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions

KEY: factor theorem, rational zero theorem

37. ANS:

By the factor theorem, x + a is a factor of P(x) if P(–a) = 0.

[pic]

PTS: 1 DIF: 2 REF: Knowledge and Understanding; Thinking

OBJ: Section 2.2 LOC: C3.2 TOP: Polynomial and Rational Functions

KEY: factor theorem

38. ANS:

Rewrite in the form P(x) = 0.

x3 + 4x2 + x – 6 = 0

Since –2 is a root, x + 2 is a factor of P(x). Divide to find the other factor.

x3 + 4x2 + x – 6 = (x + 2)(x2 + 2x – 3)

= (x + 2)(x + 3)(x – 1)

Solve x3 + 4x2 + x – 6 = 0.

(x + 2)(x + 3)(x – 1) = 0

x + 2 = 0 or x + 3 = 0 or x – 1 = 0

x = –2 or x = –3 or x = 1

Thus, the other roots are –3 and 1.

PTS: 1 DIF: 3 REF: Knowledge and Understanding; Application

OBJ: Section 2.3 LOC: C3.4 TOP: Polynomial and Rational Functions

KEY: real roots, polynomial equation

39. ANS:

Let x represent the side length of the base. Then, V(x) = x2(x + 4).

Solve x2(x + 4) = 225.

x2(x + 4) – 225 = 0

x3 + 4x2 – 225 = 0

Factor the corresponding polynomial function. Use the integral zero theorem to determine the possible values of b are ±1, ±3, ±5, ±9, ±15, ±25, ±45, ±75, and ±225. Test only positive values since x represents a side length.

Since x = 5 is a zero of the function, x – 5 is a factor. Divide to determine the other factor.

x3 + 4x2 – 225 = (x – 5)(x2 + 9x + 45)

Solve (x – 5)(x2 + 9x + 45) = 0.

x – 5 = 0 or x2 + 9x + 45 = 0

x = 5 or no solution

The dimensions of the box are 5 cm by 5 cm by 9 cm.

PTS: 1 DIF: 3 REF: Knowledge and Understanding; Thinking; Application

OBJ: Section 2.3 LOC: C3.7 TOP: Polynomial and Rational Functions

KEY: polynomial equation, real roots

40. ANS:

Let x represent the increase in each dimension. Then V(x) = (1 + x)(2 + x)(4 + x).

Solve (1 + x)(2 + x)(4 + x) = 9(1)(2)(4).

x3 + 7x2 + 14x + 8 = 72

x3 + 7x2 + 14x – 64 = 0

Factor the corresponding polynomial function using the factor theorem.

x3 + 7x2 + 14x – 64 = (x – 2)(x2 + 9x + 32)

Solve (x – 2)(x2 + 9x + 32) = 0.

x – 2 = 0 or x2 + 9x + 32 = 0

x = 2 or no solution

Each dimension should be increased by 2 m.

PTS: 1 DIF: 3 REF: Knowledge and Understanding; Thinking; Application

OBJ: Section 2.3 LOC: C3.7 TOP: Polynomial and Rational Functions

KEY: real roots, polynomial equation

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