### Problem questions on polynomials (Algebra):

If (2x – 1) is a factor of both 6x^{2} + ax – 4 and bx^{2} – 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

The remainder has no term in x^{2}, when x^{5} + kx^{2 }is divided by (x – 1) (x – 2) (x – 3), then the value of k is

(a) 90

(b) 80

(c) –80

(d) –90

Related: MCQ on Sets Relations and Functions

If x + 1 is a factor of x^{n} + 1 then the value of n is

(a) even

(b) odd

(c) any integer

(d) none

The quadratic function in x, which when divided by (x – (a), (x – (b) and (x – (c) leaves remainders 1, 2 and 4 respectively, is

(a)1/2 x^{2} – 1/2x + 1

(b) x^{2} – 1/2x + 2

(c) 2x^{2} + 1/2x + 1

(d) None of these

If 3x^{5} + 11x^{4} + 90x^{2} – 19x + 53 is divided by x + 5 then the remainder is

(a) 102

(b) 104

(c) –102

(d) –104

If x^{2} – 3x + 2 is one of the zero of x^{4} – px^{2} + 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

Related: Probability Question and Answer

The remainder when x^{4} – y^{4} is divided by (x – y) is

(a) 0

(b) x + y

(c) x^{2} – y^{2}

(d) 2y^{2
}

The remainder, when x^{3} – 7x^{2}a + 8xa^{2} + 15a^{3} is divided by (x + 2(a) is

(a) –35a^{3}

(b) –37a^{3}

(c) 37a^{3}

(d) 35 a^{3
}

The remainder, when 1 – x – x^{n} + x^{n + 1}, is divided by 1 – 2x + x^{2} is

(a) 1

(b) 2

(c) 5

(d) zero

If ax^{2} + bxy + cy^{2} is a homogeneous expression of 2nd degree where 4a = b = c and a is a square of 2 then the expression is

(a) 16x^{2} + 16xy + 4y^{2}

(b) 16x^{2} + 4xy + 16y^{2}

(c) 4x^{2} + 16xy + 16y^{2}

(d) 16x^{2} + 16xy + 16y^{2}

The multiple of 4x^{3} + 4x^{2} – x – 1 in the following is

(a) 2x + 1

(b) 2x + 3

(c) x – 1

(d) x – 2

If f(x) = x^{4} – 2x^{3} + 3x^{2} – 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

Without actual substitution, the value of x^{6} – 19x^{5} + 69x^{4} – 151x^{3} + 229x^{2} + 166x + 26, when x = 15 is

(a) 41

(b) 30

(c) 15

(d) 40

Related: Permutation examples with answers

Without actual substitution, the value of 32x^{5} – 48x^{4} + 40x^{3} – 60x^{2} + 26x – 38, when x = 1.5 is

(a) 0

(b) 2

(c) 1.5

(d) 1

If (x + (a) is a factor of x^{3} + ax^{2} – 2x + a + 4. Then the value of a is

(a) –1/3

(b) –2/3

(c) 1

(d) –4/3

Without actual division, x^{4} + 2x^{3} –2x^{2} + 2x – 3 is exactly divisible by

(a) x^{2} + 2x – 3

(b) x^{2} – 2x – 3

(c) x^{2} – 2x – 3

(d) x^{2} – 2x + 3

What must be subtracted from 4x^{4} – 2x^{3} – 6x^{2} + x – 5, so that the result is exactly divisible by 2x^{2} + x – 1 is

(a) – 5

(b) – 3

(c) – 6

(d) – 8

The degree of the polynomial (2x^{5} + 3x^{4} + 2x^{2} – 10x + 1)3 is

(a) 23

(b) 5

(c) 10

(d) 15

Related: Number Series Questions

Given that x^{2} – 4x + 1 is a factor of x4 – 9x^{3} + 27x^{2} – 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)

Sum of the values of the polynomial p(x) = x^{4} – 3x^{2} + 2x – 1 at x = 1 and x = 2 is,

(a) 4

(b) 5

(c) 6

(d) 7

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

The number that must be added to x^{3} – 3x^{2} – 12x + 19 so that 2 is a zero of the polynomial, is

(a) 6

(b) 8

(c) 9

(d) 12

Which of the following numbers is a multiple of zero of the polynomial x^{3} – x^{2} – x – 2 ?

(a) 125

(b) 27

(c) 16

(d) None of these

If ‘–a’ is a zero of the polynomial x^{n} + a^{n}, then n is

(a) even

(b) odd

(c) A or B

(d) can’t say

Related: Mensuration Problems

Given that x^{2} – 4x + 5 is a factor of x^{4} – 6x^{3} + 18x^{2} – 30x + 25, then the remaining factor is

(a) x^{2} – 2x + 5

(b) x^{2} + 2x – 5

(c) x^{2} + 2x + 5

(d) None of these

If A, B, C are the remainders of x^{3} – 3x^{2} – x + 5, 3x^{4} – x^{3} + 2x^{2} – 2x – 4, 2x^{5} – 3x^{4} + 5x^{3} – 7x^{2} + 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

If the quotient of x^{4}– 11x^{3} + 44x^{2} – 76x + 48, when divided by x^{2} – 7x + 12 is Ax^{2} + 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

If (x + 1), (x – 1), (x – 3) are the factors of x^{3} + Ax^{2} + 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

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

The expression ax^{2} + 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

Related: Ratio and Proportions formulas questions

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) 2x^{2} – 5x + 4

(b) 2x^{2} + 5x – 4

(c) 2x^{2} + 5x + 4

(d) None of these

Given that x2 – 6x + 7 is a factor of x^{3} – 11x^{2} + 37x – 35, then the other factor is

(a) (x – 2)

(b) (x – 3)

(c) (x – 1)

(d) (x – 5)

If ‘–2’ is a zero of the polynomial ax^{3} + bx^{2} + 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

If p = x^{2} + xy + 2y^{2}, Q = x^{3} + xy^{2} + y^{3}, R = a^{3} + b^{3} + c^{3}, S = a^{4} + 3a^{3}b + 4a^{2}b^{2} are homogeneous expressions then the complete homogenous expression is

(a) R

(b) Q

(c) P

(d) S

If A = x^{3} + 4x^{2}y + 5xy^{2} + y^{3}, B = x^{3} + y^{3} + x^{2} + y^{2} + x + y, C = a^{2}(b + c) + b^{2}(c + (a) + c^{2}(a^{2} + b^{3}), D = 2x^{2} + 3y^{3} + x^{3} + y^{2} + 1 are algebraic expressions, then symmetric expression is

(a) A

(b) B

(c) C

(d) D

2(x^{2} + y^{2}) + 3(x + y) + 7 is

(a) homogeneous

(b) symmetric

(c) not symmetric

(d) homogeneous symmetric

If 2x^{3} + 3x^{2} + 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

Related: LCM examples problems

What must be added to 3x^{3} + x^{2} – 22x + 9 so that the result is exactly divisible by 3x^{2} + 7x – 6 is

(a) 2x – 3

(b) 2x – 1

(c) 2x + 3

(d) 2x + 1

What must be subtracted from x^{3} – 6×2 – 15x + 80, so that the result is exactly divisible by x^{2} + x – 12 is

(a) 4(x – 1)

(b) 4(x + 1)

(c) 4(x + 2)

(d) 4(x – 2)

If x^{2} – 4 is a factor of ax^{4} + 2x^{3} – 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

If (x + (a) is a factor of x^{4} – a^{2}x^{2} + 3x – a, then the value of a is

(a) 1

(b) 2

(c) 3

(d) 0

Shalom.

Very cool. We’ve got some out Howard Heights I could show you…

Thanks Riaz â€“ great to see you back here again! ðŸ™‚