Which one of the following is a polynomial?
\(\dfrac{x^2}{2} - \dfrac{2}{x^2}\)
\(\sqrt{2x} - 1\)
\(x^2 + \dfrac{3x^{3/2}}{\sqrt{x}}\)
\(\dfrac{x-1}{x+1}\)
\(\sqrt{2}\) is a polynomial of degree
2
0
1
\(\dfrac{1}{2}\)
Degree of the polynomial \(4x^4 + 0x^3 + 0x^5 + 5x + 7\) is
4
5
3
7
Degree of the zero polynomial is
0
1
Any natural number
Not defined
If \(p(x)=x^2-2\sqrt{2}\,x+1\), then \(p(2\sqrt{2})\) is equal to
0
1
\(4\sqrt{2}\)
\(8\sqrt{2}+1\)
The value of the polynomial \(5x-4x^2+3\), when \(x=-1\), is
-6
6
2
-2
If \(p(x)=x+3\), then \(p(x)+p(-x)\) is equal to
3
\(2x\)
0
6
Zero of the zero polynomial is
0
1
Any real number
Not defined
Zero of the polynomial \(p(x)=2x+5\) is
\(-\dfrac{2}{5}\)
\(-\dfrac{5}{2}\)
\(\dfrac{2}{5}\)
\(\dfrac{5}{2}\)
One of the zeroes of the polynomial \(2x^2+7x-4\) is
2
\(\dfrac{1}{2}\)
\(-\dfrac{1}{2}\)
-2
If \(x^{51}+51\) is divided by \(x+1\), the remainder is
0
1
49
50
If \(x+1\) is a factor of the polynomial \(2x^2+kx\), then the value of \(k\) is
-3
4
2
-2
\(x+1\) is a factor of the polynomial
\(x^3+x^2-x+1\)
\(x^3+x^2+x+1\)
\(x^4+x^3+x^2+1\)
\(x^4+3x^3+3x^2+x+1\)
One of the factors of \((25x^2-1)+(1+5x)^2\) is
\(5+x\)
\(5-x\)
\(5x-1\)
\(10x\)
The value of \(249^2-248^2\) is
\(1^2\)
477
487
497
The factorisation of \(4x^2+8x+3\) is
\((x+1)(x+3)\)
\((2x+1)(2x+3)\)
\((2x+2)(2x+5)\)
\((2x-1)(2x-3)\)
Which of the following is a factor of \((x+y)^3-(x^3+y^3)\)?
\(x^2+y^2+2xy\)
\(x^2+y^2-xy\)
\(xy^2\)
\(3xy\)
The coefficient of \(x\) in the expansion of \((x+3)^3\) is
1
9
18
27
If \(\dfrac{x}{y}+\dfrac{y}{x}=-1\) \((x, y \ne 0)\), the value of \(x^3-y^3\) is
1
-1
0
\(\dfrac{1}{2}\)
If \(49x^2-b=(7x+\dfrac{1}{2})(7x-\dfrac{1}{2})\), then the value of \(b\) is
0
\(\dfrac{1}{\sqrt{2}}\)
\(\dfrac{1}{4}\)
\(\dfrac{1}{2}\)
If \(a+b+c=0\), then \(a^3+b^3+c^3\) is equal to
0
\(abc\)
\(3abc\)
\(2abc\)
Which of the following expressions are polynomials? Justify your answer:
(i) \(8\)
(ii) \(\sqrt{3}x^2 - 2x\)
(iii) \(1 - \sqrt{5}x\)
(iv) \(\dfrac{1}{5x^{-2}} + 5x + 7\)
(v) \(\dfrac{(x-2)(x-4)}{x}\)
(vi) \(\dfrac{1}{x+1}\)
(vii) \(\dfrac{1}{7}a^3 - \dfrac{2}{\sqrt{3}}a^2 + 4a - 7\)
(viii) \(\dfrac{1}{2x}\)
Polynomials: (i), (ii), (iv), (vii) because the exponent of the variable after simplification in each of these is a whole number.
Write whether the following statements are True or False. Justify your answer.
(i) A binomial can have at most two terms
(ii) Every polynomial is a binomial
(iii) A binomial may have degree 5
(iv) Zero of a polynomial is always 0
(v) A polynomial cannot have more than one zero
(vi) The degree of the sum of two polynomials each of degree 5 is always 5.
(i) False, because a binomial has exactly two terms.
(ii) False, \(x^3 + x + 1\) is a polynomial but not a binomial.
(iii) True, because a binomial is a polynomial whose degree is a whole number \(\ge 1\), so degree can be 5 also.
(iv) False, because zero of a polynomial can be any real number.
(v) False, a polynomial can have any number of zeroes. It depends upon the degree of the polynomial.
(vi) False, \(x^5 + 1\) and \(-x^5 + 2x + 3\) are two polynomials of degree 5 but the degree of the sum of the two polynomials is 1.
Classify the following polynomials as polynomials in one variable, two variables etc.:
(i) \(x^2 + x + 1\)
(ii) \(y^3 - 5y\)
(iii) \(xy + yz + zx\)
(iv) \(x^2 - 2xy + y^2 + 1\)
(i) One variable
(ii) One variable
(iii) Three variable
(iv) Two variables
Determine the degree of each of the following polynomials:
(i) \(2x - 1\)
(ii) \(-10\)
(iii) \(x^3 - 9x + 3x^5\)
(iv) \(y^3(1 - y^4)\)
(i) 1
(ii) 0
(iii) 5
(iv) 7
For the polynomial \(\dfrac{x^3 + 2x + 1}{5} - \dfrac{7}{2}x^2 - x^6\), write:
(i) the degree of the polynomial
(ii) the coefficient of \(x^3\)
(iii) the coefficient of \(x^6\)
(iv) the constant term
(i) 6
(ii) \(\dfrac{1}{5}\)
(iii) \(-1\)
(iv) \(\dfrac{1}{5}\)
Write the coefficient of \(x^2\) in each of the following:
(i) \(\dfrac{\pi}{6}x + x^2 - 1\)
(ii) \(3x - 5\)
(iii) \((x - 1)(3x - 4)\)
(iv) \((2x - 5)(2x^2 - 3x + 1)\)
(i) 1
(ii) 0
(iii) 3
(iv) \(-16\)
Classify the following as a constant, linear, quadratic and cubic polynomials:
(i) \(2 - x^2 + x^3\)
(ii) \(3x^3\)
(iii) \(5t - \sqrt{7}\)
(iv) \(4 - 5y^2\)
(v) \(3\)
(vi) \(2 + x\)
(vii) \(y^3 - y\)
(viii) \(1 + x + x^2\)
(ix) \(t^2\)
(x) \(\sqrt{2}x - 1\)
Constant Polynomial : (v)
Linear Polynomials : (iii), (vi), (x)
Quadratic Polynomials : (iv), (viii), (ix)
Cubic Polynomials : (i), (ii), (vii)
Give an example of a polynomial, which is:
(i) monomial of degree 1
(ii) binomial of degree 20
(iii) trinomial of degree 2
(i) \(10x\)
(ii) \(x^{20} + 1\)
(iii) \(2x^2 - x - 1\)
Find the value of the polynomial \(3x^3 - 4x^2 + 7x - 5\), when \(x = 3\) and also when \(x = -3\).
61, \(-143\)
If \(p(x) = x^2 - 4x + 3\), evaluate: \(p(2) - p(-1) + p\left(\dfrac{1}{2}\right)\).
\(-\dfrac{31}{4}\)
Find \(p(0)\), \(p(1)\), \(p(-2)\) for the following polynomials:
(i) \(p(x) = 10x - 4x^2 - 3\)
(ii) \(p(y) = (y + 2)(y - 2)\)
(i) \(-3, 3, -39\)
(ii) \(-4, -3, 0\)
Verify whether the following are True or False:
(i) \(-3\) is a zero of \(x - 3\)
(ii) \(-\dfrac{1}{3}\) is a zero of \(3x + 1\)
(iii) \(-\dfrac{4}{5}\) is a zero of \(4 - 5y\)
(iv) 0 and 2 are the zeroes of \(t^2 - 2t\)
(v) \(-3\) is a zero of \(y^2 + y - 6\)
(i) False
(ii) True
(iii) False
(iv) True
(v) True
Find the zeroes of the polynomial in each of the following:
(i) \(p(x) = x - 4\)
(ii) \(g(x) = 3 - 6x\)
(iii) \(q(x) = 2x - 7\)
(iv) \(h(y) = 2y\)
(i) 4
(ii) \(\dfrac{1}{2}\)
(iii) \(\dfrac{7}{2}\)
(iv) 0
Find the zeroes of the polynomial: \(p(x) = (x - 2)^2 - (x + 2)^2\).
0
By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial: \(x^4 + 1\); \(x - 1\).
\(x^3 + x^2 + x + 1\), 2
By Remainder Theorem find the remainder, when \(p(x)\) is divided by \(g(x)\), where:
(i) \(p(x) = x^3 - 2x^2 - 4x - 1\), \(g(x) = x + 1\)
(ii) \(p(x) = x^3 - 3x^2 + 4x + 50\), \(g(x) = x - 3\)
(iii) \(p(x) = 4x^3 - 12x^2 + 14x - 3\), \(g(x) = 2x - 1\)
(iv) \(p(x) = x^3 - 6x^2 + 2x - 4\), \(g(x) = 1 - \dfrac{3}{2}x\)
(i) 0
(ii) 62
(iii) \(\dfrac{3}{2}\)
(iv) \(-\dfrac{136}{27}\)
Check whether \(p(x)\) is a multiple of \(g(x)\) or not:
(i) \(p(x) = x^3 - 5x^2 + 4x - 3\), \(g(x) = x - 2\)
(ii) \(p(x) = 2x^3 - 11x^2 - 4x + 5\), \(g(x) = 2x + 1\)
(i) No
(ii) No
Show that:
(i) \(x + 3\) is a factor of \(69 + 11x - x^2 + x^3\)
(ii) \(2x - 3\) is a factor of \(x + 2x^3 - 9x^2 + 12\)
Determine which of the following polynomials has \(x - 2\) a factor:
(i) \(3x^2 + 6x - 24\)
(ii) \(4x^2 + x - 2\)
(i)
Show that \(p - 1\) is a factor of \(p^{10} - 1\) and also of \(p^{11} - 1\).
For what value of \(m\) is \(x^3 - 2mx^2 + 16\) divisible by \(x + 2\)?
1
If \(x + 2a\) is a factor of \(x^5 - 4a^2x^3 + 2x + 2a + 3\), find \(a\).
\(\dfrac{3}{2}\)
Find the value of \(m\) so that \(2x - 1\) be a factor of \(8x^4 + 4x^3 - 16x^2 + 10x + m\).
\(-2\)
If \(x + 1\) is a factor of \(ax^3 + x^2 - 2x + 4a - 9\), find the value of \(a\).
2
Factorise:
(i) \(x^2 + 9x + 18\)
(ii) \(6x^2 + 7x - 3\)
(iii) \(2x^2 - 7x - 15\)
(iv) \(84 - 2r - 2r^2\)
(i) \((x + 6)(x + 3)\)
(ii) \((3x - 1)(2x + 3)\)
(iii) \((x - 5)(2x + 3)\)
(iv) \(2(7 + r)(6 - r)\)
Factorise:
(i) \(2x^3 - 3x^2 - 17x + 30\)
(ii) \(x^3 - 6x^2 + 11x - 6\)
(iii) \(x^3 + x^2 - 4x - 4\)
(iv) \(3x^3 - x^2 - 3x + 1\)
(i) \((x - 2)(x + 3)(2x - 5)\)
(ii) \((x - 1)(x - 2)(x - 3)\)
(iii) \((x + 1)(x - 2)(x + 2)\)
(iv) \((x - 1)(x + 1)(3x - 1)\)
Using suitable identity, evaluate the following:
(i) \(103^3\)
(ii) \(101 \times 102\)
(iii) \(999^2\)
(i) 1092727
(ii) 10302
(iii) 998001
Factorise the following:
(i) \(4x^2 + 20x + 25\)
(ii) \(9y^2 - 66yz + 121z^2\)
(iii) \(\left(2x + \dfrac{1}{3}\right)^2 - \left(x - \dfrac{1}{2}\right)^2\)
(i) \((2x + 5)^2\)
(ii) \((3y - 11z)^2\)
(iii) \(\left(3x - \dfrac{1}{6}\right)\left(x + \dfrac{5}{6}\right)\)
Factorise the following:
(i) \(9x^2 - 12x + 3\)
(ii) \(9x^2 - 12x + 4\)
(i) \(3(x - 1)(3x - 1)\)
(ii) \((3x - 2)(3x - 2)\)
Expand the following:
(i) \((4a - b + 2c)^2\)
(ii) \((3a - 5b - c)^2\)
(iii) \((-x + 2y - 3z)^2\)
(i) \(16a^2 + b^2 + 4c^2 - 8ab - 4bc + 16ac\)
(ii) \(9a^2 + 25b^2 + c^2 - 30ab + 10bc - 6ac\)
(iii) \(x^2 + 4y^2 + 9z^2 - 4xy - 12yz + 6xz\)
Factorise the following:
(i) \(9x^2 + 4y^2 + 16z^2 + 12xy - 16yz - 24xz\)
(ii) \(25x^2 + 16y^2 + 4z^2 - 40xy + 16yz - 20xz\)
(iii) \(16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz\)
(i) \((3x + 2y - 4z)(3x + 2y - 4z)\)
(ii) \((-5x + 4y + 2z)(-5x + 4y + 2z)\)
(iii) \((4x - 2y + 3z)(4x - 2y + 3z)\)
If \(a + b + c = 9\) and \(ab + bc + ca = 26\), find \(a^2 + b^2 + c^2\).
29
Expand the following:
(i) \((3a - 2b)^3\)
(ii) \(\left(\dfrac{1}{x} + \dfrac{y}{3}\right)^3\)
(iii) \(\left(4 - \dfrac{1}{3x}\right)^3\)
(i) \(27a^3 - 54a^2b + 36ab^2 - 8b^3\)
(ii) \(\dfrac{1}{x^3} + \dfrac{y}{x^2} + \dfrac{y^2}{3x} + \dfrac{y^3}{27}\)
(iii) \(64 - \dfrac{16}{x} + \dfrac{4}{3x^2} - \dfrac{1}{27x^3}\)
Factorise the following:
(i) \(1 - 64a^3 - 12a + 48a^2\)
(ii) \(8p^3 + \dfrac{12}{5}p^2 + \dfrac{6}{25}p + \dfrac{1}{125}\)
(i) \((1 - 4a)(1 - 4a)(1 - 4a)\)
(ii) \(\left(2p + \dfrac{1}{5}\right)\left(2p + \dfrac{1}{5}\right)\left(2p + \dfrac{1}{5}\right)\)
Find the following products:
(i) \(\left(\dfrac{x}{2} + 2y\right)\left(\dfrac{x^2}{4} - xy + 4y^2\right)\)
(ii) \((x^2 - 1)(x^4 + x^2 + 1)\)
(i) \(\dfrac{x^3}{8} + 8y^3\)
(ii) \(x^6 - 1\)
Factorise:
(i) \(1 + 64x^3\)
(ii) \(a^3 - 2\sqrt{2}b^3\)
(i) \((1 + 4x)(1 - 4x + 16x^2)\)
(ii) \((a - \sqrt{2}b)(a^2 + \sqrt{2}ab + 2b^2)\)
Find the following product: \((2x - y + 3z)(4x^2 + y^2 + 9z^2 + 2xy + 3yz - 6xz)\).
\(8x^3 - y^3 + 27z^3 + 18xyz\)
Factorise:
(i) \(a^3 - 8b^3 - 64c^3 - 24abc\)
(ii) \(2\sqrt{2}a^3 + 8b^3 - 27c^3 + 18\sqrt{2}abc\)
(i) \((a - 2b - 4c)(a^2 + 4b^2 + 16c^2 + 2ab - 8bc + 4ac)\)
(ii) \((\sqrt{2}a + 2b - 3c)(2a^2 + 4b^2 + 9c^2 - 2\sqrt{2}ab + 6bc + 3\sqrt{2}ac)\)
Without actually calculating the cubes, find the value of:
(i) \(\left(\dfrac{1}{2}\right)^3 + \left(\dfrac{1}{3}\right)^3 - \left(\dfrac{5}{6}\right)^3\)
(ii) \((0.2)^3 - (0.3)^3 + (0.1)^3\)
(i) \(-\dfrac{5}{12}\)
(ii) -0.018
Without finding the cubes, factorise:
\((x - 2y)^3 + (2y - 3z)^3 + (3z - x)^3\)
\(3(x - 2y)(2y - 3z)(3z - x)\)
Find the value of:
(i) \(x^3 + y^3 - 12xy + 64\), when \(x + y = -4\)
(ii) \(x^3 - 8y^3 - 36xy - 216\), when \(x = 2y + 6\)
(i) 0
(ii) 0
Give possible expressions for the length and breadth of the rectangle whose area is given by \(4a^2 + 4a - 3\).
Length = \(2a - 1\), Breadth = \(2a + 3\)
If the polynomials \(az^3 + 4z^2 + 3z - 4\) and \(z^3 - 4z + a\) leave the same remainder when divided by \(z - 3\), find the value of \(a\).
\(-1\)
The polynomial \(p(x) = x^4 - 2x^3 + 3x^2 - ax + 3a - 7\) when divided by \(x + 1\) leaves the remainder \(19\). Find the value of \(a\). Also find the remainder when \(p(x)\) is divided by \(x + 2\).
\(a = 5\)
Remainder when divided by \(x + 2\) is \(62\).
If both \(x - 2\) and \(x - \dfrac{1}{2}\) are factors of \(px^2 + 5x + r\), show that \(p = r\).
Without actual division, prove that \(2x^4 - 5x^3 + 2x^2 - x + 2\) is divisible by \(x^2 - 3x + 2\).
Hint: Factorise \(x^2 - 3x + 2\).
Simplify \((2x - 5y)^3 - (2x + 5y)^3\).
\(-120x^2y - 250y^3\)
Multiply \(x^2 + 4y^2 + z^2 + 2xy + xz - 2yz\) by \((-z + x - 2y)\).
\(x^3 - 8y^3 - z^3 - 6xyz\)
If \(a, b, c\) are all non-zero and \(a + b + c = 0\), prove that \(\dfrac{a^2}{bc} + \dfrac{b^2}{ca} + \dfrac{c^2}{ab} = 3\).
If \(a + b + c = 5\) and \(ab + bc + ca = 10\), then prove that \(a^3 + b^3 + c^3 - 3abc = -25\).
Prove that \((a + b + c)^3 - a^3 - b^3 - c^3 = 3(a + b)(b + c)(c + a)\).