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Question (41)

QN0036561

A monic cubic polynomial has zeroes 2, -1 and 3. Its constant term is:

(A) -6
(B) 6
(C) 5
(D) -5

Correct Answer: B

Explanation:

Constant term = -αβγ = -

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Question (42)

QN0036562

If the zeroes of a cubic polynomial are -1, 2 and 5, then the sum of products of zeroes taken two at a time is:

(A) -7
(B) 3
(C) -13
(D) 7

Correct Answer: B

Explanation:

αβ + βγ + γα = (-1)(2) + (2)(5) + (5)(-1) = -2 + 10 - 5 = 3.

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Question (43)

QN0036563

A cubic polynomial has zeroes 1, -2 and 3. Which is a monic polynomial with these zeroes?

(A) x³ - 2x² - 5x + 6
(B) x³ + 2x² - 5x - 6
(C) x³ - 2x² + 5x - 6
(D) x³ + 2x² + 5x + 6

Correct Answer: A

Explanation:

(x - 1)(x + 2)(x - 3) = x³ - 2x² - 5x + 6.

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Question (44)

QN0036564

If α, β and γ are zeroes of x³ - 3x² - 4x + 12, then α + β + γ is:

(A) -3
(B) 3
(C) 4
(D) -4

Correct Answer: B

Explanation:

α + β + γ = -b/a = -(-3)/1 = 3.

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Question (45)

QN0036565

If α, β and γ are zeroes of 2x³ + 6x² - 8x - 24, then αβ + βγ + γα equals:

(A) -4
(B) 4
(C) -3
(D) 3

Correct Answer: A

Explanation:

αβ + βγ + γα = c/a = -8/2 = -4.

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Question (46)

QN0036566

A polynomial graph intersects the x-axis at four distinct points. What can be concluded about its degree at minimum?

(A) 1
(B) 2
(C) 3
(D) 4

Correct Answer: D

Explanation:

A polynomial of degree n can have at most n zeroes. Therefore, four distinct zeroes require degree at least 4.

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Question (47)

QN0036567

A polynomial of degree 5 has four distinct zeroes. Which statement is necessarily true?

(A) It has exactly four zeroes
(B) It has exactly five zeroes
(C) It can have at most one additional zero
(D) It cannot have any additional zero

Correct Answer: C

Explanation:

A degree-5 polynomial can have at most 5 zeroes. Therefore, at most one additional zero can exist.

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Question (48)

QN0036568

If α and β are zeroes of p(x) = ax² + bx + c, which expression must equal zero?

(A) a(α + β) + b
(B) aαβ + c
(C) a(α + β) - b
(D) aαβ - c

Correct Answer: A

Explanation:

Since α + β = -b/a, multiplying by a gives a(α + β) + b = 0.

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Question (49)

QN0036569

If the sum and product of the zeroes of a quadratic polynomial are -4 and 3 respectively, which polynomial can represent it?

(A) x² - 4x + 3
(B) x² + 4x + 3
(C) x² + 3x - 4
(D) x² - 3x + 4

Correct Answer: B

Explanation:

Polynomial = x² - (sum)x + product = x² + 4x + 3.

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Question (50)

QN0036570

If α, β and γ are zeroes of ax³ + bx² + cx + d, which relation is correct?

(A) α + β + γ = b/a
(B) αβ + βγ + γα = -c/a
(C) αβγ = -d/a
(D) αβγ = d/a

Correct Answer: C

Explanation:

For a cubic polynomial, α + β + γ = -b/a, αβ + βγ + γα = c/a and αβγ = -d/a.

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Question (41)

A monic cubic polynomial has zeroes 2, -1 and 3. Its constant term is:

Explanation:

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Question (42)

If the zeroes of a cubic polynomial are -1, 2 and 5, then the sum of products of zeroes taken two at a time is:

Explanation:

Link:

Question (43)

A cubic polynomial has zeroes 1, -2 and 3. Which is a monic polynomial with these zeroes?

Explanation:

Link:

Question (44)

If α, β and γ are zeroes of x³ - 3x² - 4x + 12, then α + β + γ is:

Explanation:

Link:

Question (45)

If α, β and γ are zeroes of 2x³ + 6x² - 8x - 24, then αβ + βγ + γα equals:

Explanation:

Link:

Question (46)

A polynomial graph intersects the x-axis at four distinct points. What can be concluded about its degree at minimum?

Explanation:

Link:

Question (47)

A polynomial of degree 5 has four distinct zeroes. Which statement is necessarily true?

Explanation:

Link:

Question (48)

If α and β are zeroes of p(x) = ax² + bx + c, which expression must equal zero?

Explanation:

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Question (49)

If the sum and product of the zeroes of a quadratic polynomial are -4 and 3 respectively, which polynomial can represent it?

Explanation:

Link:

Question (50)

If α, β and γ are zeroes of ax³ + bx² + cx + d, which relation is correct?

Explanation:

Link:

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