Class 9 Pearson - Polynomials and Square Roots of Algebraic Expressions

If the degree of a polynomial AB is 15 and the degree of polynomial B is 5, then the degree of polynomial A is
(1) 3
(2) 8
(3) 4
(4) 10

The expression 21x² + 11x – 2 equal to
(1) (x – 2) (7x + 1)
(2) (7x + 1) (3x – 2)
(3) (7x – 1) (3x – 1)
(4) (7x – 1) (3x + 2)

If the LCM and HCF of two polynomials are 90 m⁵a⁶b³x² and m³a⁵ respectively and also one of the monomial is 18 m⁵a⁶x², then the other monomial is
(1) 5 m³a⁵b³
(2) 15 m⁵a³b²
(3) 5 m⁵a³b⁵
(4) 15 m³a⁵b⁴

The remainder when x³ – 3x² + 5x – 1 is divided by x + 1 is ____________.
(1) –8
(2) –12
(3) –10
(4) –9

Which of the following is a homogeneous expression?
(1) 4x² – 5xy + 5x²y + 10y²
(2) 5x + 10y + 100
(3) 14x³ + 15x²y + 16y²x + 24y³
(4) x² + y² + x + y + 1

Σx(y³ – z³) = __________.
(1) (x – y) (y – z) (z – x) (x + y + z)
(2) (x – y) (y – z) (x – z) (x – y – z)
(3) (x + y) (y + z) (z + x) (x + y + z)
(4) (x + y) (y + z) (z + x) (x – y – z)

The remainder when f(x) 4x³ – 3x² + 2x – 1 is divided by 2x + 1 is ____________.
(1) 1
(2) $\frac{- 3}{4}$
(3) $\frac{- 13}{4}$
(4) $\frac{- 7}{4}$

If the HCF of the polynomials 12a³b⁴c², 18a⁴b³c³and 24a⁶b²c⁴ is ________.
(1) 12a³b²c²
(2) 12a⁶b⁴c⁴
(3) 6a³b²c²
(4) 48a⁶b⁴c⁴

Find the value of a, if (x + 2) is a factor of the polynomial f(x) = x³ + 13x² + ax + 20.
(1) –15
(2) 20
(3) 25
(4) 32

10.

The polynomial x³ – 4x² + x – 4 on factorization gives
(1) (x – 4) (x² – 1)
(2) (x – 4) (x² + 4)
(3) (x + 4) (x² + 1)
(4) (x – 4) (x² + 1)

11.

If expression ax³ + 2x²y – bxy² – 2y³is symmetric, then (a, b) =
(1) (2, 2)
(2) (–2, 2)
(3) (–2, –2)
(4) (2, –2)

12.

The square root of $y^{2} + \frac{1}{y^{2}} + 2$ is
(1) $y + \frac{1}{y}$
(2) $y - \frac{1}{y}$
(3) $y^{2} + \frac{1}{y^{2}}$
(4) $y^{2} - \frac{1}{y^{2}}$

13.

The product of the polynomial 2x³ – 3x² + 6 and x² – x is _________.
(1) 2x⁶ – 5x⁴ + 3x³+ 6x² – 6x
(2) 2x⁵ – x⁴ + 3x³– 6x² + 6x
(3) 2x⁵ – 5x⁴ + 3x³+ 6x² – 6x
(4) None of these

14.

The LCM of x² – 16 and 2x² – 9x + 4 is
(1) (2x + 1) (x + 4) (x – 4)
(2) (x² + 16) (2x + 1)
(3) 2(1 – 2x) (x + 4) (x – 4)
(4) (2x – 1) (x + 4) (x – 4)

15.

If P = 3x³ – 5x + 9, Q = 4x³ + 5x² – 11 and R = 5x³ + 4x² – 3x + 7, then P – 2Q + R is
(1) 2(3x² + 4x – 19)
(2) –6x² – 5x + 38
(3) 2(3x² + 4x + 19)
(4) – 2(3x² + 4x – 19)

16.

If g(x) = 3a^x + 7a²b – 13ab² + 9b^y is a homogeneous expression in terms of a and b, then the values of x and y respectively are _______.
(1) 2, 2
(2) 2, 1
(3) 3, 2
(4) 3, 3

17.

The polynomial $11 a^{2} - 12 \sqrt{2} a + 2$ on factorization gives
(1) $(11 a + \sqrt{2}) (a - \sqrt{2})$
(2) $(a - \sqrt{2}) (11 a - \sqrt{2})$
(3) $(a + 11) (a + \sqrt{2})$
(4) $(11 a - \sqrt{2}) (a + \sqrt{2})$

18.

If xⁿ + 1 is divisible by x + 1, n must be.
(1) any natural number
(2) an odd natural number
(3) an even natural number
(4) None of these

19.

What is the first degree expression to be subtracted from x⁶ + 8x⁴ + 2x³ + 16x² + 4x + 5 in order to make it a perfect square?
(1) –4x – 4
(2) 4x + 4
(3) 4x – 4
(4) –4x + 4

20.

Find the square root of $\frac{m^{n^{2}} n^{m^{2}} a^{(m + n)}}{{(m + n)}^{{(m + n)}^{2}}}$ .
(1) $m^{n} n^{m} a^{\frac{m + n}{2}}$
(2) $\frac{m^{\frac{n^{2}}{2}} n^{\frac{m^{2}}{2}} a^{\frac{m + n}{2}}}{{(m + n)}^{\frac{(m + n) 2}{2}}}$
(3) $\frac{m^{n} n^{m} a^{\sqrt{m + n}}}{{(m + n)}^{(m + n)}}$
(4) None of there

21.

What is the first degree expression to be added to 16x⁶ + 8x⁴– 2x³ + x² + 2x + 1 in order to make it a perfect square?
(1) $\frac{5}{2} x + \frac{15}{16}$
(2) $- \frac{5}{2} x - \frac{15}{16}$
(3) $- \frac{5}{2} x + \frac{15}{16}$
(4) $+ \frac{2}{2} x + \frac{15}{16}$

22.

Factorize the polynomial $8 x^{3} - \frac{1}{64}$
(1) $(2 x - \frac{1}{4}) (4 x^{2} - \frac{x}{2} + \frac{1}{16})$
(2) $(2 x - \frac{1}{8}) (4 x^{2} + \frac{x}{2} - 16)$
(3) $(2 x - \frac{1}{4}) (4 x^{2} + \frac{1}{16} + \frac{x}{2})$
(4) $(2 x - \frac{1}{4}) (4 x^{2} + \frac{x}{2} - 16)$

23.

The product of polynomials 3x³ – 4x² + 7 and x² +1 is
(1) 3x⁵ – 4x⁴ + 3x³ + 3x² + 7
(2) x⁵ + 4x² – 2x + 3
(3) 3x⁵ – 4x⁴ – 3x³ + 4x + 8
(4) 3x⁵ – 5x⁴ + 8x² + 2x + 1

24.

If LCM and HCF of two monomials is 60x⁴y⁵a⁶b⁶ and 5x²y³respectively. If one of the two monomials is 15x⁴y³a⁶, then the other monomial is
(1) 12x²y³a⁶b⁶
(2) 20x⁴y⁵b⁶
(3) 20x²y⁵b⁶
(4) 15x²y⁵b⁶

25.

Which of the following is a factor of the polynomial f(x) = 2x³ – 5x² + x + 2?
(1) x + 1
(2) x + 2
(3) 2x + 1
(4) 2x – 1

26.

If 3x – 1 is a factor of the polynomial 81x³ – 45x² + 3a – 6, then a is __________.
(1) $\frac{8}{3}$
(2) $\frac{- 7}{3}$
(3) $\frac{- 10}{3}$
(4) $\frac{11}{3}$

27.

If A = 4x³ – 5x + 7, B = 2x³ – x² + 3 and C = 5x³ – 8x² + 10, then A – 2B – C is
(1) 5x³ – 2x² + x + 4
(2) –5x³ + 10x² – 5x – 9
(3) x³ + 10x² – 5x + 9
(4) 5x³ – 8x² + x – 1

28.

The square root of $x^{m^{2} - n^{2}}, x^{n^{2} + 2 m n}, x^{n^{2}}$ is
(1) x^m⁺ⁿ
(2) $x^{{(m + n)}^{2}}$
(3) $x^{(m + n) / 2}$
(4) $x^{\frac{1}{2} {(m + n)}^{2}}$

29.

x⁸³¹ + y⁸³¹ is always divisible by
(1) x – y
(2) x² + y²
(3) x + y
(4) None of these

30.

If (x + 1) (x + 2) (x + 3) (x + k) + 1 is a perfect square, then the value of k is
(1) 4
(2) 5
(3) 6
(4) 7

31.

If A = 6x⁴ + 5x³ – 14x² + 2x + 2 and B = 3x² – 2x – 1, then the remainder when A ÷ B is
(1) x
(2) 2x
(3) 3x
(4) 4x

32.

The polynomial x⁵ – a²x³ – x²y³ + a²y³ on factorization gives
(1) (x – y) (x – a) (x + a) (x² – y²+ xy)
(2) (x + a) (x – y) (x – a) (x² – y²+ xy)
(3) (x + a) (x – y) (x – a) (x²+ y²+ xy)
(4) None of these

33.

The HCF of the polynomials x⁴ + 6x² + 25, x³ – 63x² + 7x – 5 and x² + 5 – 2x is
(1) x²– 2x – 5
(2) x²– 2x + 5
(3) x – 1
(4) 3x² + 2

34.

The HCF of the polynomials (2x – 1) (5x² – ax + 3) and (x – 3) (2x² + x + b) is (2x –1) (x – 3). Then the values of a and b respectively are _______.
(1) 16, –1
(2) –16, 1
(3) –16, –1
(4) 16, 1

35.

The remainder when x⁴⁵ is divided by x² – 1 is
(1) 2x
(2) –x
(3) 0
(4) x

36.

Factorize $\sum_{a, b, c} a^{2} (b^{4} - c^{4})$ .
(1) (a – b)² (b – c)² (c – a)²
(2) (a – b) (a + b) (b – c) (b + c) (c – a) (c + a)
(3) (a + b)² (b + c)² (c + a)²
(4) None of these

37.

The polynomial 6y⁴ – 19y² + 23y² + 10y + 8 on factorization gives
(1) (y + 1) (y – 4) (3y + 2) (2y + 1)
(2) (y + 1) (y – 4) (3y – 2) (2y – 1)
(3) (y + 1) (y – 4) (3y – 2) (2y + 1)
(4) (y + 1) (y – 4) (3y + 2) (2y – 1)

38.

If the LCM of the polynomials (y – 3)^a (2y + 1)^b (y + 13)⁷ and (y – 3)⁴ (2y +1)⁹ (y + 13)^c is (y – 3)⁶ (2y +1)¹⁰ (y + 13)¹⁷, then the least value of a + b + c is
(1) 23
(2) 3
(3) 10
(4) 16

39.

If the LCM of the polynomials 195(x + 3)² (x – 2) (x + 1)² and 221(x + 1)³ (x +3) (x + 4) is __________.
(1) 221(x + 3)² (x – 1)² (x – 2) (x – 14)
(2) 13(x + 3) (x + 1)²
(3) 3315(x + 3)² (x + 1)³ (x – 2) (x + 4)
(4) None of these

40.

The what value of k the HCF of x² + x + (5k – 1) and x² – 6x + (3k +11) is (x – 2)?
(1) 2
(2) 2
(3) – 2
(4) – 1

41.

The HCF of the polynomials 9(x + a)^p (x – b)^q (x + c)^r and 12(x + a)^p^{+ 3} (x – b)^q^{– 3} (x + c)^r^{+ 2} is 3(x + a)⁶ (x – b)⁶ (x + c)⁶ , then the value of p + q – r is
(1) 21
(2) 9
(3) 15
(4) 6

42.

The remainders obtained when the polynomial x³ + x² – 9x – 9 divided by x, x + 1 and x + 2 respectively are __________.
(1) – 9, 0, – 15
(2) – 9, – 16, 5
(3) 0, 0, 5
(4) – 9, 0, 5

43.

Find the value of $\frac{{(a + b)}^{2}}{(b - c) (c - a)} + \frac{{(b + c)}^{2}}{(a - b) (c - a)} + \frac{{(c + a)}^{2}}{(a - b) (b - c)}$
(1) – 1
(2) 0
(3) 1
(4) 2

44.

Find the square root of the expression $\frac{1}{x y z} (x^{2} + y^{2} + z^{2}) + 2 (\frac{1}{x} + \frac{1}{y} + \frac{1}{z})$
(1) $\frac{x + y + z}{x y z}$
(2) $\sqrt{\frac{y z}{x}} + \sqrt{\frac{z x}{y}} + \sqrt{\frac{x y}{z}}$
(3) $\sqrt{x} + \sqrt{y} + \sqrt{z}$
(4) $\sqrt{\frac{x}{y z}} + \sqrt{\frac{y}{x z}} + \sqrt{\frac{z}{x y}}$

45.

Factorize the expression $9 x^{4} + \frac{1}{x^{4}} + 2$
(1) $(3 x^{2} - \frac{1}{x^{2}} + 2) (3 x^{2} + \frac{1}{x^{2}} + 2)$
(2) $(3 x^{2} - \frac{1}{x^{2}} - 2) (3 x^{2} + \frac{1}{x^{2}} + 2)$
(3) $(3 x^{2} - \frac{1}{x^{2}} + 2) (3 x^{2} - \frac{1}{x^{2}} + 2)$
(4) $(3 x^{2} + \frac{1}{x^{2}} + 2) (3 x^{2} + \frac{1}{x^{2}} - 2)$

46.

The following are the steps involved in factorizing 64x⁶ – y⁶. Arrange them in sequential order.
(A) {(2x)³ + y³}{(2x)³ – y³}
(B) (8x³)² – (y³)²
(C) (8x³ + y³) (8x³ – y³)
(D) (2x + y) (4x² – 2xy + y²) (2x – y) (4x² + 2xy + y²)
(1) BADC
(2) BDAC
(3) BCAD
(4) BACD

47.

If a + b + c = 0, show that a³ + b³ + c³ = 3abc. The following are the steps involved in showing the above result. Arrange them in sequential order.
(A) a³ + b³ + 3ab(–c) = –c³
(B) (a + b)³ = (–c)³
(C) a³ + b³ + 3ab(a + c) = –c³
(D) a³ + b³ + c³ = 3abc
(1) ABDCE
(2) BCDAE
(3) CBDAE
(4) CADBE

48.

If the HCF of 8x³ y^a and 12x^by² is 4x^a y^b, then find the maximum value of a + b.
(1) 2
(2) 4
(3) 6
(4) Cannot be determined

49.

The polynomial 5x⁵ – 3x³ + 2x² – k gives a remainder 1, when divided by x + 1. Find the value of k.
(1) 5
(2) –1
(3) 2
(4) 1

50.

Factorize: a³ + b³ + 3abc – 1.
(1) (a + b – 1) (a² + b² + a + b + 1 – ab)
(2) (a + b – 1) (a² + b² + a + b – 1 + ab)
(3) (a + b – 1) (a² + b² – a – b + 1 + ab)
(4) None of these

51.

If f and g are two polynomials of degrees 3 and 4 respectively, then what is the degree of f – g?
(1) 1
(2) 3
(3) 4
(4) Cannot be determined

52.

Find the square root of $\frac{x^{2}}{9} + \frac{9}{4 x^{2}} - \frac{x}{3} - \frac{3}{2 x} + \frac{5}{4}$ .
(1) $\frac{2 x}{3} + \frac{3}{2 x} - \frac{1}{2}$
(2) $\frac{x}{3} - \frac{3}{2 x} + 1$
(3) $\frac{3}{x} + \frac{2}{3 x} - \frac{1}{2}$
(4) $\frac{x}{3} + \frac{3}{2 x} - \frac{1}{2}$

53.

The Square root of (xy + xz – yz)² – 4xyz (x – y) is ____________.
(1) xy – yz – 2xyz
(2) (x + y – 2xy)
(3) (xy + 3 – y)
(4) ( xy + yz –zx)

54.

$(\sum_{x, y, z} {(x + 1)}^{2}) - {(\sum_{x, y, z} (x))}^{2} - 3 =$ _________.
(1) $2 [\sum_{x, y, z} x - \sum_{x, y, z} x y]$
(2) $3 [\sum_{x, y, z} x^{2} - \sum_{x, y, z} x]$
(3) $2 [\sum_{x, y, z} x y - \sum_{x, y, z} x^{2}]$
(4) $3 [\sum_{x, y, z} x^{2} - \sum_{x, y, z} x]$

55.

${(\sum_{x, y, z} x)}^{2} - (\sum_{x, y, z} x^{2})$ = ______.
(1) $\sum_{x, y, z} x$
(2) $2 (\sum_{x, y, z} x y)$
(3) $\underset{x, y, z}{π} x y$
(4) $2 (\sum_{x, y, z} x + y)$

56.

If $\sqrt{4 x^{4} + 12 x^{3} + 25 x^{2} + 24 x + 16} = a x^{2} + b x + c$ , then which of he following is true?
(1) 2b = a – c
(2) 2a = b + c
(3) 2b = a + c
(4) 2x = c – a

57.

Find the square root of the algebraic expression which is the average of the following expressions $x^{2} + \frac{1}{x^{2}}, - 2 (x - \frac{1}{x}) and - 1.$
(1) $\frac{x}{\sqrt{3}} - \frac{1}{\sqrt{3}} + \frac{1}{x}$
(2) $\frac{x}{\sqrt{3}} + 1 + x$
(3) $\frac{1}{\sqrt{3}} (1 - x - \frac{1}{x})$
(4) None of these

58.

Find the square root of $\frac{a^{2}}{4} + \frac{1}{a^{2}} - \frac{1}{a} + \frac{a}{2} - \frac{3}{4} .$
(1) $\frac{a}{2} - \frac{1}{a} + \frac{1}{2}$
(2) $\frac{a}{2} + \frac{2}{a} - 1$
(3) $\frac{a}{2} + \frac{1}{a} - \frac{1}{2}$
(4) $\frac{a}{2} - \frac{2}{a} - \frac{1}{2}$

59.

$\frac{{(x + y)}^{3} + {(x - y)}^{3}}{2} - y (3 x^{2} + y^{2}) =$ _________.
(1) x³– y³
(2) (x – y)³
(3) 2x³– 3x²y
(4) x³– 6xy²

60.

Find the square root of (4a + 5b + 5c)² – (5a + 4b + 4c)² + 9a².
(1) ^{$\sqrt{3} (b + c)$}
(2) 3(b + c – a)
(3) 3(b + c)
(4) 3(b + c – a)

61.

$\frac{{(a - b)}^{3} - {(a + b)}^{3}}{2} + a (a^{2} + 3 b^{2}) =$ __________.
(1) a³ – b³
(2) (a + b)³
(3) a³ + b³
(4) (a – b)³

62.

The square root of (3a + 2b + 3c)² – (2a + 3b + 2c)² + 5b² is
(1) $\sqrt{5} (a + b + c)$
(2) $\sqrt{5} (a + b)$
(3) $\sqrt{5} (a + c)$
(4) $\sqrt{5} (a + b - c)$

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