Question Set D

CBSE 9th Standard Mathematics Polynomials Sample Question Papers Study Material - QB365

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Questions:

Multiple Choice Question

95 x 1 = 95

Product of \((x-\frac{1}{x})(x+\frac{1}{x})(x^2+\frac{1}{x^2})\) is:

(a)

\(x^4+\frac{1}{x^4}\)

(b)

\(x^3+\frac{1}{x^3}-2\)

(c)

\(x^4-\frac{1}{x^4}\)

(d)

\(x^2+\frac{1}{x^2}+2\)

If the area of a rectangle is \(4x^2+4x-3,\) then it's possible dimensions are:

(a)

2x-3, 2x+1

(b)

2x-1, 2x+3

(c)

3x+1, 2x-3

(d)

3x-1, 2x+3

If a+b+c=5 and ab+bc+ca=10, then the value of \(a^3+b^3+c^3-3abc\) is:

(a)

(b)

-25

(c)

(d)

\(4a^2+9b^2+c^2+12ab+4ac+6bc\) is:

(a)

\((2a+3b+c)^2\)

(b)

\((a+2b+3c)^2\)

(c)

\((2a+b+3c)^2\)

(d)

\((3a+b+2c)^2\)

\(x^3+8y^3+6x^2y+12xy^2 \) is:

(a)

\((x+2y)^3\)

(b)

\((2x+y)^3\)

(c)

\((x+y)^3\)

(d)

\((x+3y)^3\)

The expanded form of \((x+2y+z)^2\) is:

(a)

\(x^2+4y^2+z^2+4xy+4yz+2zx\)

(b)

\(x^2+4y^2+z^2+2xy+2yz+zx\)

(c)

\(x^2+4y^2+z^2+4xy+2yz+4zx\)

(d)

\(x^2+4y^2+z^2+4xy+4yz+4zx\)

The expanded form of \((x-y-z)^2\) is:

(a)

\(x^2+y^2+z^2-2xy+2yz-2zx\)

(b)

\(x^2+y^2+z^2-xy-yz-zx\)

(c)

\(x^2+y^2+z^2+2xy-2yz-2zx\)

(d)

\(x^2+y^2+z^2+ 2xy+2yz+2zx\)

The expanded form of \((x+\frac{1}{3})^3\) is:

(a)

\(x^3+\frac{1}{27}+\frac{x}{3}+x^2\)

(b)

\(x^3+\frac{1}{9}+\frac{x}{3}+x^3\)

(c)

\(x^3+\frac{1}{27}+\frac{x^2}{3}+x\)

(d)

\(x^3+\frac{1}{27}+3x+3x^2\)

One of the factors of \((x^3-1)-(x-1)\) is:

(a)

\(x+1\)

(b)

\(x^2-1\)

(c)

\(x-1\)

(d)

\(x+4\)

10.

One of the factors of \(42+y-y^2\) is:

(a)

\((7+y)\)

(b)

\((6-y)\)

(c)

\((7-y)\)

(d)

\((-6+y)\)

11.

One of the factors of \((1+3y)^2+ (9y^2-1)\) is:

(a)

\((1-3y)\)

(b)

\((3-y)\)

(c)

\((3y+1)\)

(d)

\((y-3)\)

12.

One of the factors of \((16y^2-1)+(1-4y)^2\) is:

(a)

\((4+y)\)

(b)

\((4-y)\)

(c)

\((4y+1)\)

(d)

\(8y\)

13.

If \(\frac{x}{y}+\frac{y}{x}=-1(x,y\neq0)\) the value of \(x^3-y^3\) is:

(a)

(b)

-1

(c)

\(\frac{1}{2}\)

(d)

14.

If \(a+b=1\) then the value of \(a^3+b^3+3ab\) is

(a)

(b)

-1

(c)

(d)

-2

15.

If \(x+\frac{1}{x}=4\) then the value of \(x^2+\frac{1}{x^2}\) is:

(a)

(b)

(c)

(d)

16.

If \(x+y=9\) and \(xy=20,\) value of \(x^2+y^2\) is:

(a)

(b)

(c)

(d)

141

17.

If \(2(a^2+b^2)=(a+b)^2\) then

(a)

a=2b

(b)

b=2a

(c)

a=b

(d)

a+b=0

18.

The factor of \((2a-b)^3+(b-2c)^3+8(c-a)^3\) is:

(a)

(2a-b)(b-2c)(c-a)

(b)

3(2a-b)(b-2c)(c-a)

(c)

6(6a-b)(b-2c)(c-a)

(d)

2a x b x 2c

19.

If \(x^{\frac{1}{3}}+y^{\frac{1}{3}}+z^{\frac{1}{3}}=0\) then which one of the following expressions is correct?

(a)

\(x^3+y^3+z^3=0\)

(b)

\(x+y+z=3x^{\frac{1}{3}}+y^{\frac{1}{3}}+z^{\frac{1}{3}}\)

(c)

\(x+y+z=3xyz\)

(d)

\(x^3+y^3+z^3=3xyz\)

20.

If a+b+c=0, then \(a^3+b^3+c^3\) is equal to:

(a)

abc

(b)

-3abc

(c)

(d)

3abc

21.

The value of 549²-548² is:

(a)

1087

(b)

1077

(c)

1097

(d)

22.

Value of 525²-475² is:

(a)

100

(b)

10000

(c)

50000

(d)

100000

23.

The factors of \(x^3+9x^2+23x+15\) are

(a)

(x+1)(x+3)(x+5)

(b)

(x+1)(x+3)(x-5)

(c)

(x+1)(x-3)(x-5)

(d)

(x-1)(x-3)(x-5)

24.

For what value of p, is the polynomial \(2x^4+3x^3+2px^2+3x+6\) divisible by x+2?

(a)

(b)

(c)

(d)

-1

25.

(x+2) is a factor of \(2x^3+5x^2-x-k.\) The value of k is:

(a)

(b)

-24

(c)

-6

(d)

26.

If (x+3) is the factor of polynomial \(x^3+ax^2+x+3\) then the value of a is:

(a)

(b)

(c)

(d)

-3

27.

For what value of a, is the polynomial \(x^3+2x^2-3ax-8\) divisible by x-4?

(a)

\(\frac{22}{3}\)

(b)

\(\frac{11}{3}\)

(c)

(d)

28.

For what value of b, is the polynomial \(x^3-3x^2+bx-6\) divisible by x-3?

(a)

(b)

(c)

(d)

-3

29.

The value of p for which x+p is a factor of \(x^2+px+3-p \) is:

(a)

(b)

-1

(c)

(d)

-3

30.

If x-2 is a factor of \(5x^2-kx-18,\) then the value of k is:

(a)

-1

(b)

(c)

(d)

31.

In which of the following (x+2) is a factor?

(a)

\(4x^3-13x+6\)

(b)

\(x^3+x^2+x+4\)

(c)

\(4x^3+13x-25\)

(d)

\(-2x^3+x^2-x-19\)

32.

If \(x^2+ky+6=(x+2)(x+3)\) for all x, the value of k is

(a)

(b)

-1

(c)

(d)

33.

The factors of \(a^7+ab^6\) are:

(a)

\(a, (a^6+b^6)\)

(b)

\(b, (a^6+b^6)\)

(c)

\(a^6,(a+b)\)

(d)

\(b^6, (a+b)\)

34.

If \(x^{11}+101 \) is divided by (x+1), the remainder is:

(a)

-1

(b)

102

(c)

(d)

100

35.

If \(x^{31}+51 \) is divided by (x+1), the remainder is:

(a)

(b)

(c)

(d)

36.

Find the remainder when the polynomial \(2x^3+13x^2+x-70\) is divided by x-2.

(a)

(b)

-2

(c)

(d)

-70

37.

What is remainder when \(x^3-2x^2+x+1\) is divided by (x-1)?

(a)

(b)

-1

(c)

(d)

38.

The remainder when \(x^2+2x+1\)is divided by (x+1) is:

(a)

(b)

(c)

(d)

-2

39.

On dividing \(5y^3-2y^2-7y+1\) by y, the remainder we get is:

(a)

-1

(b)

(c)

(d)

40.

The remainder obtained when the polynomial p(x) is divided by (b-ax) is:

(a)

\(p(-\frac{b}{a})\)

(b)

\(p(\frac{a}{b})\)

(c)

\(p(\frac{b}{a})\)

(d)

\(p(-\frac{a}{b})\)

41.

When p(x) is divided by ax-b, then the remainder is:

(a)

p(a+b)

(b)

\(p(-\frac{b}{a})\)

(c)

\(p(\frac{a}{b})\)

(d)

\(p(\frac{b}{a})\)

42.

If the polynomial p(x) is divided by (x+3), then the remainder will be:

(a)

p(-1)

(b)

p(2)

(c)

p(-3)

(d)

p(3)

43.

If a polynomial f(x) is divided by x-a, then remainder is:

(a)

f(0)

(b)

f(a)

(c)

f(-a)

(d)

f(a)-f(0)

44.

A zero of \(2x^3-7x^2-16x+5\) is

(a)

(b)

(c)

(d)

-1

45.

The zeros of the polynomial \(x^2+2 x+3\) are

(a)

real

(b)

not real

(c)

irrational

(d)

rational

46.

If \(p(x)=x^2-2\sqrt{2}x+1,\) then \(p(2\sqrt{2})\) is:

(a)

(b)

(c)

\(4\sqrt{2}\)

(d)

\(3\sqrt{2}+1\)

47.

If \(p(x)=x^3+x^2+\sqrt{5}x+\sqrt{5}\) then the value of \(p(-\sqrt{5})\) is:

(a)

\(-5\sqrt{5}\)

(b)

\(-4\sqrt{5}\)

(c)

\(5+\sqrt{5}\)

(d)

\(-5+\sqrt{5}\)

48.

If (2t+1) is the factor of the polynomial \(p(t)=4t^3+4t^2-t-1,\) then the value of \(p(-\frac{1}{2})\)

(a)

\(-\frac{1}{2}\)

(b)

\(\frac{1}{2}\)

(c)

(d)

49.

Find \(p(\frac{1}{3})\) for \(p(t)=t^2-t+2\)

(a)

\(\frac{22}{9}\)

(b)

\(\frac{14}{9}\)

(c)

\(\frac{16}{9}\)

(d)

\(\frac{15}{9}\)

50.

If \(p(x)=3x-7,\) then \(p(x)+p(-x)\) is:

(a)

(b)

(c)

(d)

-14

51.

If \(p(x)=x^3-x^2+x+1\) then value of p(1)+p(-1) is:

(a)

\(\frac{1}{4}\)

(b)

(c)

(d)

-2

52.

If \(p(x)=2+\frac{x}{2}+x^2-\frac{x^2}{3},\) then p(-1) is:

(a)

\(\frac{15}{6}\)

(b)

\(\frac{17}{6}\)

(c)

\(\frac{1}{6}\)

(d)

\(\frac{13}{6}\)

53.

If \(p(x)=7-3x+2x^2\) then value of p(-2) is:

(a)

(b)

(c)

(d)

54.

The value of the polynomial \(x^2-x-1\) at x=-1 is:

(a)

-3

(b)

(c)

-1

(d)

55.

The value of polynomial \(6a^2+7a-3\) when a=1 is:

(a)

(b)

(c)

-13

(d)

-4

56.

The zeros of \(f(x)=x^2+2x\) are:

(a)

0, -2

(b)

1, 2

(c)

0, 2

(d)

1, -2

57.

The zeros of the polynomial p(x)=(x-6)(x-5) are:

(a)

-6, -5

(b)

-6, 5

(c)

6, -5

(d)

6, 5

58.

Which of the following polynomial has -3 as a zero?

(a)

(x-3)

(b)

\(x^2-9\)

(c)

\(x^2-3x\)

(d)

\(x^2+3\)

59.

Zero of the polynomial p(x) = cx + d is:

(a)

-d

(b)

-c

(c)

\(\frac{d}{c}\)

(d)

\(-\frac{d}{c}\)

60.

The maximum number of zeros of the polynomial p(y)=my^a is:

(a)

a+1

(b)

(c)

m+1

(d)

61.

Zero of the polynomial p(x) when p(x) = ax, a ≠ 0 is:

(a)

(b)

(c)

(d)

\(\frac{1}{a}\)

62.

Zero of the zero polynomial is:

(a)

(b)

(c)

Any real number

(d)

Not defined

63.

The coefficient of \(x^2\)in \((3x+x^3)(x+\frac{1}{x })\)

(a)

(b)

(c)

(d)

64.

The coefficient of x² in \((3x^2-5)(4+4x^2)\) is:

(a)

-12

(b)

(c)

-8

(d)

65.

The coefficient of x in the product of (x-1)(1-2x) is:

(a)

-3

(b)

(c)

-2

(d)

66.

The coefficient of x in the expression of \((x+2)^3\) is:

(a)

(b)

(c)

(d)

67.

The coefficient of x² in the polynomial \(7+4x-x^2+x^3\) is:

(a)

-1

(b)

(c)

(d)

68.

In the polynomial \(1-\sqrt{11} x,\) the coefficient of x is:

(a)

(b)

(c)

\(-\sqrt{11}\)

(d)

\(\sqrt{11}\)

69.

The maximum number of terms in a polynomial of degree 10 is:

(a)

(b)

(c)

(d)

70.

If p=17, the degree of the polynomial \(p(x)=(p-x)^3+14\) is:

(a)

(b)

(c)

(d)

71.

The degree of the polynomial \(p(x)=(x-7)^3-x^3\) is:

(a)

(b)

(c)

(d)

72.

The degree of the polynomial \((x^3+5)(4-x^5)\) is:

(a)

(b)

(c)

(d)

73.

Degree of polynomial \((x^3+5)(4-x^5)\) is:

(a)

(b)

(c)

(d)

74.

The degree of the polynomial \(2-y^2-y^3+2y^7\) is:

(a)

(b)

(c)

(d)

75.

Degree of the polynomial \(p(x)=4x^4+2x^3+x^5+2x+7\) is:

(a)

(b)

(c)

(d)

76.

Degree of the polynomial \(4x^4+0x^3+0x^5+5x+7\) is:

(a)

(b)

(c)

(d)

77.

A cubic polynomial is a polynomial with degree:

(a)

(b)

(c)

(d)

78.

Degree of which of the following polynomials is zero?

(a)

(b)

(c)

(d)

\(x+\frac{1}{x}\)

79.

The degree of the polynomial p(x)=3 is:

(a)

(b)

(c)

(d)

80.

\(\sqrt{2}\) is an example of:

(a)

(b)

(c)

(d)

\(\frac{1}{2}\)

81.

\((1+3x)^3\) is an example of:

(a)

Monomial

(b)

Binomial

(c)

Trinomial

(d)

None of these

82.

Which of the following is cubic polynomial?

(a)

\(x^3+3x^2-4x+3\)

(b)

\(x^2+4x-7\)

(c)

\(3x^2+4\)

(d)

\(3(x^2+x+1)\)

83.

Which of the following is a trinomial in x?

(a)

\(x^3+1\)

(b)

\(x^3+x^2+x\)

(c)

\(x\sqrt{x}-\sqrt {x}+1\)

(d)

\(x^3+2x\)

84.

Which of the following is a binomial in y?

(a)

\(y^2+2\)

(b)

\(y+\frac{1}{y}+2\)

(c)

\(\sqrt{y}+\sqrt{2}y\)

(d)

\(y\sqrt{y}+1\)

85.

Which of the following is a quadratic polynomial in one variable?

(a)

\(\sqrt{2x^3}+5\)

(b)

\(2x^2+2x^-2\)

(c)

\(x^2\)

(d)

\(2x^2+y^2\)

86.

A cubic polynomial has number of zeroes:

(a)

(b)

(c)

(d)

At least three

87.

A linear polynomial

(a)

has one and only one zero

(b)

may have no zero

(c)

may have one zero

(d)

may have more than one zero

88.

Which of the following is a polynomial in one variable?

(a)

\(3-x^2+x\)

(b)

\(\sqrt{3x}+4\)

(c)

\(x^3+y^3+7\)

(d)

\(x+\frac{1}{x}\)

89.

\(y+\frac{1}{y}\) is:

(a)

polynomial of degree 1

(b)

polynomial of degree 2

(c)

polynomial of degree 3

(d)

Not a polynomial

90.

Which of the following is a polynomial?
In (a), negative power of x; in (b) and (c), fractional powers of x

(a)

\(x^2+x+\frac{3}{x^2}\)

(b)

\(\sqrt{x}+5\)

(c)

\(x^{3/4}-7x+4\)

(d)

\(\frac{3}{2}x^3-\frac{4}{3}x^2+2x-1\)

91.

The number 0 is called a

(a)

zero polynomial

(b)

binomial

(c)

trinomial

(d)

linear polynomial

92.

Select the correct statement from the following:

(a)

Degree of a zero polynomial is zero.

(b)

Degree of a zero polynomial is not defined.

(c)

Degree of a constant polynomial is not defined

(d)

Zero of the polynomial is not defined

93.

The compact form of (x+y)(x-y) is
\((x+y)(x-y)=x^2-y^2\) is an algebraic identity

(a)

\(x^2+y^2\)

(b)

\(x^2-2xy+y^2\)

(c)

\(x^2+2xy+y^2\)

(d)

\(x^2-y^2\)

94.

The expansion for \((x-y)^2\) is
\((x-y)^2=x^2-2xy+y^2\) is an algebraic identity

(a)

\(x^2-2xy+y^2\)

(b)

\(x^2+2xy+y^2\)

(c)

\(x^2+y^2\)

(d)

\(x^2-y^2\)

95.

Which of the following is an algebraic identity?
\((x+y)^2=x^2+2xy+y^2\) is an algebraic identity

(a)

\((x+y)^2=x^2-2xy+y^2\)

(b)

\((x+y)^2=x^2+2xy-y^2\)

(c)

\((x+y)^2=x^2+2xy+y^2\)

(d)

\((x+y)^2=-x^2+2xy+y^2\)

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Answers Key:

Multiple Choice Question

95 x 1 = 95

\((x-\frac{1}{x})(x+\frac{1}{x})(x^2+\frac{1}{x^2})\)
\((x^2-\frac{1}{x^2})(x^2+\frac{1}{x^2})=x^4-\frac{1}{x^4}\)

\((2x-1)(2x+3)=4x^2+4x-3\)

\(a^3+b^3+c^3-3abc\)
\(=(a+b+c)(z^2+b^2+c^2-ab-bc-ca)\)
\(=(a+b+c)(a+b+c)^2-3(ab+bc+ca)\)
\(=5(5)^2-3(10)\)
\(=-25\)

\(4a^2+9b^2+c^2+12ab+4ac+6bc\)
\(=(2a)^2+(3b)^2+(c)^2+2(2a)(3b)+2(3c)(c)+2(c)(2a)\)
\((2a+3b+c)^2\)

\(x^3+8y^3+6x^2y+12xy^2 \)
\(=(x)^3+(2y)^3+3(x)(2y)(x+2y)\)
\(=(x+2y)^3\)

\((x+2y+z)^2=(x)^2+(2y)^2+(z)^2+2(x)(2y)+2(2y)(z)+2(z)(x)\)

\((x-y-z)^2=[x+(-y)+(-z)]^2\)
\(=(x)^2+(-y)^2+(-z)^2\) \(+2(x)(-y)+2(-y)(-z)+2(-z)(x)\)

\((x+\frac{1}{3})^3=x^3+(\frac{1}{3})^3+3x.\frac{1}{3}(x+\frac{1}{3})\)

\(x^3-1=(x-1)(x^2+x+1)\)

10.

\(42+y-y^2=42+7y-6y-y^2\)
\(=7(6+y)-y(6+y)\)
\(=(6+y)(7-y)\)

11.

\((1+3y)^3+(9y^2-1)\)
\(=(1+3y)^2+(3y+1)(3y-1)\)
\(=(1+3y)(1+3y+3y-1)\)
\(=6(1+3y)y\)

12.

\((16y^2-1)+(1-4y)^2\)
\(=16y^2-1+1+16y^2-8y\)
\(=8y(4y-1)\)
Aliter: \((16y^2-1)+(1-4y)^2\)
\(=(4y-1)(4y+1)+(4y-1)^2\)
\(=(4y-1)(4y+1+4y-1)\)
\((4y-1)8y\)

13.

\(\frac{x}{y}+\frac{y}{x}=-1\ \Rightarrow\ x^2+y^2+xy=0 \)
\(\therefore x^3-y^3=(x-y)(x^2+y^2+xy)=0\)

14.

\(a^3+b^3+3ab\)
\(=a^3+b^3+3ab(a+b) \ \because \ a+b=0\)
\(=(a+b)^2=1^3=1\)

15.

\((x+\frac{1}{x})^2=x^2+\frac{1}{x^2}+2\)
\(\Rightarrow (4)^2=x^2+\frac{1}{x^2}+2\)
\(\Rightarrow x^2+\frac{1}{x^2}=14\)

16.

\((x+2)^2=x^2+y^2+2xy\)
\(\Rightarrow (9)^2=x^2+y^2+2(20)\)
\(\Rightarrow \ x^2+y^2=41\)

17.

\(2a^2+2b^2=a^2+b^2+2ab\)
\(\Rightarrow \ a^2+b^2-2ab=0\)
\(\Rightarrow\ (a-b)^2=0\)
\(\Rightarrow\ a-b=0 \Rightarrow\ a=b\)

18.

If a+b+c=0 then \(a^3+b^3+c^3=3abc\)

19.

If a+b+c=0 then \(a^3+b^3+c^3=3abc\)

20.

(d)
3abc

21.

549²-548²
=(549+548)(549-548)=1097

22.

525²-475²
=(525+475)(525-475)=50000

23.

\(f(x)=x^3+9x^2+23x+15\)
\(f(-1)=0\)
\(f(-3)=0\)
\(f(-5)=0\)

24.

\(f(x)=2x^4+3x^3+2px^2+3x+6\)
\(f(-2)=0\)
\(\Rightarrow \ 2(-2)^4+3(-2)^3+2p(-2)^2+3(-2)+6=0\)
\(\Rightarrow \ 32-24+8p=0\)
\(\Rightarrow \ p=-1\)

25.

Use factor theorem

26.

\(f(x)=x^3+ax^2+x+3\)
\(x+3=0\Rightarrow \ x=-3\)
\(f(-3)=0\)
\(\Rightarrow \ (-3)^3+a(-3)^2+(-3)+3=0\)
\(\Rightarrow \ a=0\)

27.

\(f(x)=x^3+2x^2-3ax-8\)
\(\Rightarrow \ f(4)=0\)
\(\Rightarrow \ 4^3+2 (4)^2-3a(4)-8=0\)
\(\Rightarrow \ 64+32-12a-8=0\)
\(\Rightarrow \ 12a=88\)
\(\Rightarrow \ a=\frac{22}{3}\)

28.

\(f(x)=x^3+3x^2+bx-6\)
\(f(3)=0\)
\(\Rightarrow \ 3^3-3 \times3^2+b\times 3-6=0\)
\(\Rightarrow\ b=2\)

29.

\(x+p=0\ \Rightarrow \ x=-p\)
By factor theorem,
\((-p)^2+p(-p)+3-p=0 \ \Rightarrow \ p=3\)

30.

\(x-2=0\quad \Rightarrow \quad x=2\)
\(f(x)=5x^2-kx-18\)
\(f(2)=0\)
\(\Rightarrow \ 5(2)^2-k(2)-18=0\)
\(\Rightarrow \ k=1\)

31.

Use factor theorem

32.

\(x^2+kx+6=(x+2)(x+3)=x^2+5x+6\)
\(\Rightarrow \ k=5\)

33.

\(a^7+ab^6\)
\(=a(a^6+b^6)\)

34.

\(x+1=0\ \ Rightarrow \ x=-1\)
\(\therefore\) Remainder=\((-1)^11+101=-1+101\)
=100

35.

By remainder theorem,
Remainder\(=(-1)^{31}+51=-1+51\)
\(=50 \quad |x+1=0\ \Rightarrow \ x=-1\)

36.

\(f(x)=2x^3+13x^2+x-70\)
ஃ Remainder=\(f(2)=2(2)^3+13(2)^2+2-70\)
\(=16+52+2-70=0\)

37.

\(x-1=0\ \Rightarrow\ x=1\)
Remainder=\((1)^3-2(1)^2+(1)+1=1\)
By remainder theorem

38.

Remainder \(=f(-1)=(-1)^2+2(-1)+1=0\)
Aliter: \(x^2+2x+1=(x+1)^2\)
ஃ Remainder =0

39.

y=y-0
ஃ Remainder \(=5(0)^3-2(0)^2-7(0)+1=1\)

40.

Use remainder theorem
\(b-ax=0\ \Rightarrow\ x=\frac{b}{a}\)

41.

\(ax-b=0\ \Rightarrow\ x=\frac{b}{a}\)
Use remainder theorem

42.

x+3=0
\(\Rightarrow\) x=-3
\(\therefore\) Remainder=p(-3)

43.

Remainder theorem

44.

\(p(x)=2x^3-7x^2-16x+5\)
\(\therefore\quad p(5)=2(5)^3-7(5)^2-16(5)+5\)
\(=250-175-80+5=0\)

45.

\(x^2+2 x+3=0\)
\(\Rightarrow\quad x=\frac{-2\pm\sqrt{4-12}}{2}=\frac{-2\pm2\sqrt{2}i}{2}\)
\(=-1\pm\sqrt{2}i\)

46.

\(p(x)=x^2-2\sqrt{2}x+1\)
\(\therefore\) \(p(2\sqrt{2})=(2\sqrt{2})^2-(2\sqrt{2})(2\sqrt{2})+1\)
= 0

47.

\(p(x)=x^3+x^2+\sqrt{5}x+\sqrt{5}\)
\(\therefore\) \(p(-\sqrt5)=(-\sqrt{5})^3+(-\sqrt{5})^2+\sqrt{5}(-\sqrt{5})+\sqrt{5}\)
\(=-5\sqrt{5}+5-5+\sqrt{5}=-4\sqrt{5}\)

48.

\(p(t)=4t^3+4t^2-t-1\)
\(\because\quad p(-\frac{1}{2})\)
\(=4(-\frac{1}{2})^3+4(-\frac{1}{2})^2-(-\frac{1}{2})-1\)
\(=-\frac{1}{2}+1+\frac{1}{2}=0\)

49.

\(p(\frac{1}{3})=(\frac{1}{3})^2-(\frac{1}{3})+2=\frac{16}{9}\)

50.

Evaluate p(-x)

51.

Evaluate p(-1) and p(1)

52.

\(p(-1)=2+\frac{-1}{2}+(-1)^2-\frac{(-1)^3}{3}=\frac{17}{6}\)

53.

\(p(-2)=7-3(-2)+2(-2)^2=21\)

54.

Value\(=(-1)^2-(-1)-1=1\)

55.

Value=\(6(1)^2+7(1)-3\)
\(=6+7-3=10\)

56.

\(f(x)=0 \quad \Rightarrow \quad x^2+2x+0\)
\(\Rightarrow \quad x(x+2)=0\quad \Rightarrow\quad x=0,-12\)

57.

\((x-6)(x-5)=0\quad \Rightarrow\quad\quad x=6, 5\)

58.

\(x^2-9=(x-3)(x+3) \quad |\quad x-(-3)=x+3\)

59.

\(cx+d=0\ \Rightarrow\ x=-\frac{d}{c}\)

60.

Evident

61.

\(ax=0\ \Rightarrow\ x=0\)

62.

By convention

63.

Coefficient of \(x^2=3+1=4\)

64.

Coefficient of \(x^2=(3)(4)+(-5)(4)=-8\)

65.

Coefficient of x=1+2=3

66.

\((x+2)^3=x^3+8+6x^2+12x\)

67.

The terms containing x² is-x² , i.e., (-1) x²

68.

Evident

69.

A polynomial of degree n has maximum number of terms as (n+1)

70.

\(p(x)=(p-x)^3+14=(17-x)^3+14\)
\(=(17)^3-x^3-3(17)^2(x)+3.17.x^2+14\)
\(\because\) Degree=3

71.

\((x-7)^3-x^3=-21x^2+147x-343\)

72.

\((x^3+5)(4-x^5)\)\(=4x^3-x^8+20-5x^5\)

73.

Highest power of x=3+2=5

74.

Highest power of y=7

75.

Highest power of x=5

76.

Highest power of x=4

77.

Definition of cubic polynomial

78.

15 is a constant polynomial

79.

The degree of a non-zero constant polynomial is zero.

80.

\(\sqrt{2}\) is a constant polynomial

81.

\((1+3x)^3\) \(=1+27x^3+9x+27x^2\) It has 4 terms

82.

Degree of \(x^3+3x^2-4x+3\) is 3

83.

\(x^3+x^2+x\) has three terms

84.

\(y^2+2\) has two terms; three terms in (b) fractional power of y in (c) and (d);

85.

Definition of quadratic polynomial

86.

By definition

87.

A characteristic of a linear polynomial

88.

Fractional power of x in (b),
Two variables in (c),
Negative power of x in (d)

89.

\(\frac{1}{y}=y^-1\) has negative exponent so, not a polynomial

90.

(d)
\(\frac{3}{2}x^3-\frac{4}{3}x^2+2x-1\)

91.

Definition of zero polynomial

92.

Convention

93.

(a)
\(x^2+y^2\)

94.

(a)
\(x^2-2xy+y^2\)

95.

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