CAT sample papers

Polynomials Questions and Answers

Polynomials Algebra

Problem questions on polynomials (Algebra):

Ques. If (2x – 1) is a factor of both 6x2 + ax – 4 and bx2 – 11x + 3, then the values of a and b are
(A) a = 5, b = 10
(b) a = 3, b = 10
(c) b = 2a, a = 4
(d) a = 2b, b = 2
Ans. (a)

Ques. The remainder contains no term in x2, when x5 + kx2 is divided by (x – 1) (x – 2) (x – 3), then the value of  k is
(A) 90
(b) 80
(c) –80
(d) –90
Ans. (d)

Related: MCQ on Sets Relations and Functions

Ques. If x + 1 is a factor of xn + 1 then the value of n is
(A) even
(b) odd
(c) any integer
(d) none
Ans. (b)

Ques. If 3x5 + 11x4 + 90x2 – 19x + 53 is divided by x + 5 then the remainder is
(A) 102
(b) 104
(c) –102
(d) –104
Ans. (c)

Ques. If x2 – 3x + 2 is one of the zero of x4 – px2 + q then the values of p and q are
(A) 4 and 6
(b) 3 and 5
(c) 5 and 6
(d) 5 and 4
Ans. (d)

Related: Probability Question and Answer

Ques. The remainder when x4 – y4 is divided by (x – y) is
(A) 0
(b) x + y
(c) x2 – y2
(d) 2y2
Ans. (a)

Ques. The remainder when x3 – 7x2a + 8xa2 + 15a3 is divided by (x + 2(a) is
(A) –35a3
(b) –37a3
(c) 37a3
(d) 35 a3
Ans. (b)

Ques. The remainder when 1 – x – xn + xn + 1,  is divided by 1 – 2x + x2 is
(A) 1
(b) 2
(c) 5
(d) zero
Ans. (d)

Ques. If ax2 + bxy + cy2 is a homogeneous expression of 2nd degree where 4a = b = c and a is a square of 2 then the expression is
(A) 16x2 + 16xy + 4y2
(b) 16x2 + 4xy + 16y2
(c) 4x2 + 16xy + 16y2
(d) 16x2 + 16xy + 16y2
Ans. (c)

Ques. The multiple of 4x3 + 4x2 – x – 1 in the following is
(A) 2x + 1
(b) 2x + 3
(c) x – 1
(d) x – 2
Ans. (a)

Ques. If f(x) = x4 – 2x3 + 3x2 – ax + b is a polynomial such that when it is divided by x – 1 and x + 1, the remainders are respectively 5 and 19. Then the remainder when f(x) is divided by (x – 2) is
(A) 15
(b) 10
(c) 20
(d) 5
Ans. (b)

Ques. Without actual substitution, the value of  x6 – 19x5 + 69x4 – 151x3 + 229x2 + 166x + 26, when x = 15 is
(A) 41
(b) 30
(c) 15
(d) 40
Ans. (a)

Related: Permutation examples with answers

Ques. Without actual substitution, the value of 32x5 – 48x4 + 40x3 – 60x2 + 26x – 38, when x  = 1.5 is
(A) 0
(b) 2
(c) 1.5
(d) 1
Ans. (d)

Ques. If (x + (a) is a factor of x3 + ax2 – 2x + a + 4. Then the value of a is
(A) –1/3
(b) –2/3
(c) 1
(d) –4/3
Ans. (d)

Ques. Without actual division, x4 + 2x3 –2x2 + 2x – 3 is exactly divisible by
(A) x2 + 2x – 3
(b) x2 – 2x – 3
(c) x2 – 2x – 3
(d) x2 – 2x + 3
Ans. (a)

Ques. What must be subtracted from 4x4 – 2x3 – 6x2 + x – 5, so that the result is exactly divisible by 2x2 + x – 1 is
(A) – 5
(b) – 3
(c) – 6
(d) – 8
Ans. (c)

Ques.  The degree of the polynomial (2x5 +  3x4 + 2x2 – 10x + 1)3 is
((a) 23
(b) 5
(c) 10
(d) 15
Ans. (d)

Related: Number Series Questions

Ques. Given that x2 – 4x + 1 is a factor of x4 – 9x3 + 27x2 – 29x + 6, then the other factors are
(A) (x + 2), (x – 3)
(b) (x – 2), (x + 3)
(c) (x – 2), (x – 3)
(d) (x + 2), (x + 3)
Ans. (c)

Ques.  Sum of the values of the polynomial p(x) = x4 – 3x2 + 2x – 1 at x = 1 and x = 2 is,
(A) 4
(b) 5
(c) 6
(d) 7
Ans. (c)

Ques. If a monomial, a binomial and a trinomial are added, then the minimum and maximum number of terms in the obtained polynomial are respectively
(A) 1, 3
(b) 3, 6
(c) 1, 6
(d) 3, 3
Ans. (c)

Ques. The number, that must be added to x3 – 3x2 – 12x  + 19 so that 2 is a zero of the polynomial, is
(A) 6
(b) 8
(c) 9
(d) 12
Ans. (c)

Ques. Which of the following number is a multiple of the zero of the polynomial x3 – x2 – x – 2 ?
(A) 125
(b) 27
(c) 16
(d) None of these
Ans. (c)

Ques. If ‘–a’ is a zero of the polynomial xn + an, then n is
(A) even
(b) odd
(c) A or B
(d) can’t say
Ans. (b)

Related: Mensuration Problems

Ques. Given that x2 – 4x + 5 is a factor of x4 – 6x3 + 18x2 – 30x + 25, then the remaining factor is
(A) x2 – 2x + 5
(b) x2 + 2x – 5
(c) x2 + 2x + 5
(d) None of these
Ans. (a)

Ques. If A, B, C are the remainders of x3 – 3x2 – x + 5, 3x4 – x3 + 2x2  – 2x – 4, 2x5 – 3x4 + 5x3 – 7x2 + 3x – 4, when divided by (x + 1), (x + 2), (x – 2) respectively, then the ascending order of A, B, C is
(A) A, B, C
(b) B, C, A
(c) A, C, B
(d) B, A, C
Ans. (c)

Ques. If the quotient of x4– 11x3 + 44x2 – 76x + 48, when divided by x2 – 7x + 12 is Ax2 + Bx + C, then the descending order of A, B, C is
(A) A, B, C
(b) B, C, A
(c) A, C, B
(d) C, A, B
Ans. (d)

Ques. If (x + 1), (x – 1), (x – 3) are the factors of x3 + Ax2 + Bx + C, then the ascending order of A, B, C is
(A) A, B, C
(b) B, C, A
(c) A, C, B
(d) B, A, C
Ans. (a)

Ques. If ax + by is a homogenous expression of first degree in x and y. When x = 1, y = 2 the value of expression is 9 and when x = 2, y = 3 the value of equation is 16. Then the expression is
(A) 2x + 5y
(b) 5x + 2y
(c) 5x + 4y
(d) 5x + 3y
Ans. (b)

Ques. The expression ax2 + bx + c equals –2 when x = 0, leaves remainder 3 when divided by (x – 1) and a remainder –3 when divided by (x + 1). Then the values of a, b, c are respectively
(A) 2, 3
(b) –2, 3
(c) 2, –3
(d) –2, –3
Ans. (a)

Related: Ratio and Proportions formulas questions

Ques. The quadratic polynomial in x, which when divided by (x – 1), (x – 2) and (x – 3) leaves remainders of 11, 22 and 37 is
(A) 2x2 – 5x + 4
(b) 2x2 + 5x – 4
(c)  2x2 + 5x + 4
(d)  None of these
Ans. (c)

Ques. Given that x2 – 6x + 7 is a factor of x3 – 11x2 + 37x – 35, then the other factor is
(A) (x – 2)
(b) (x – 3)
(c) (x – 1)
(d) (x – 5)
Ans. (d)

Ques. If ‘–2’ is a zero of the polynomial ax3 + bx2 + x – b and the value of the polynomial is 4 at x = 2, then the values of a and b respectively are
(A) – ¼, 0
(b)  – ½, 0
(c) – ½, ½
(d) – ¼, ½
Ans. (a)

Ques. If p = x2 + xy + 2y2, Q = x3 + xy2 + y3, R = a3 + b3 + c3, S = a4 + 3a3b + 4a2b2 are homogeneous expressions then the complete homogenous expression is
(A) R
(b) Q
(c) P
(d) S
Ans. (a)

Ques. If A = x3 + 4x2y + 5xy2 + y3, B = x3 + y3 + x2 + y2 + x + y, C = a2(b + c) + b2(c + (a) + c2(a2 + b3), D = 2x2 + 3y3 + x3 + y2 + 1 are algebraic expressions, then symmetric expression is
(A) A
(b) B
(c) C
(d) D
Ans. (b)

Ques. 2(x2 + y2) + 3(x + y) + 7 is
(A) homogeneous
(b) symmetric
(c) not symmetric
(d) homogeneous symmetric
Ans. (b)

Ques. If 2x3 + 3x2 + ax + b is divided by ( x – 2) leaves remainder 2 and divided by (x + 2) leaves remainder –2, then the values of a, b are respectively
(A) 7, 12
(b) –7, 12
(c) 7, –12
(d) –7, –12
Ans. (d)

Related: LCM examples problems

Ques. What must be added to 3x3 + x2 – 22x + 9 so that the result is exactly divisible by 3x2 + 7x – 6 is
(A) 2x – 3
(b) 2x – 1
(c) 2x + 3
(d)  2x + 1
Ans. (c)

Ques. What must be subtracted from x3 – 6×2 – 15x + 80, so that the result is exactly divisible by x2 + x – 12 is
(A) 4(x – 1)
(b) 4(x + 1)
(c) 4(x + 2)
(d) 4(x – 2)
Ans. (a)

Ques. If x2 – 4 is a factor of ax4 + 2x3 – 3×2 + bx – 4, then the values of a and b are
(A) a = 1, b = – 8
(b) a = 3, b = 5
(c) a = 6, b = 7
(d) a = – 1, b = 5
Ans. (a)

Ques. If (x + (a) is a factor of x4 – a2x2 + 3x – a, then the value of a is
(A) 1
(b) 2
(c) 3
(d) 0
Ans. (d)

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Vishal Arora

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