If a + b + c =0, then a3+b3 + c3 is equal to
Zero of the polynomial p(x) = 2x + 5 is
If x51+51 is divided by x + 1, the remainder is..
If x + 1 is a factor of the polynomial 2x2 + kx, then the value of k is
One of the factors of (25x2 – 1) + (1 + 5x)2 is
The value of 2492 – 2482 is
x + 1 is a factor of the polynomial
If 49x^2-b=\left(7x+\frac{1}{2}\right)\left(7x-\frac{1}{2}\right), then the value of b is
Which of the following is a factor of (x+ y)3 – (x3 + y3)?
The coefficient of x in the expansion of (x + 3)3 is
If \frac{x}{y}+\frac{y}{x}=-1(\text{ where }x,y\ne0), the value of x3+y3 is
One of the zeroes of the polynomial 2x2 + 7x –4 is
The factorization of 4x2 + 8x+ 3 is
Which one of the following is a polynomial?
√2 is a polynomial of degree
What is the degree of the polynomial 4x^4+0x^3+0x^5+5x+7?
Degree of the zero polynomial is
If p(x)= x2 – 2√2x + 1, then p(2√2) is equal to
The value of the polynomial 5x – 4x2 + 3, when x = – 1 is
If p(x) = x + 3, then p(x) + p(–x) is equal to
Zero of the zero polynomial is
Which one of the following is a polynomial ?
(a) \frac{x^{2}}{2}-\frac{2}{x^{2}}
(b) \sqrt{2 x}-1
(c) x^{2}+\frac{3 x^{\frac{3}{2}}}{\sqrt{x}}
(d) \frac{x-1}{x+1}
(A) 2
(B) 0
(C) 1
(D) ½
4×4 +0x³ + 0x5 + 5x+ 7 is
(A) 4
(B) 5
(C) 3
(d)7
(A) 0
(B) 1
(C) Any natural number
(D) Not defined
If p(x)= x² – 2√2x + 1, then p(2√2) is equal to
(C) 4√2
(D) 8√2 +1
The value of the polynomial 5x – 4x² + 3, when x = – 1 is
(A) – 6
(B) 6
(C) 2
(D) – 2
(A) 3
(B) 2x
(C) 0
(D) 6
(C) Any real number
(A) – 2/5
(B) – 5/2
(C) 2/5
(D) 5/2
One of the zeroes of the polynomial 2x² + 7x –4 is
(B) ½
(C) – ½
(D) -2
Write whether the following statement is true or false:
Every polynomial is a Binomial.
Find whether \sqrt{3}x^2-2x is a polynomial?
A Binomial can have atmost two terms.
Find whether \frac{1}{2\mathrm{x}} is a polynomial?
Find whether \frac{1}{7}a^3-\frac{2}{\sqrt{3}}a^2+4a-7 is a polynomial?
Find whether \frac{1}{x+1} is a polynomial?
Find whether 1-\sqrt{5x} is a polynomial?
Find whether \frac{1}{5x^{-2}}+5x+7 is a polynomial?
The degree of the sum of two polynomials each of degree 5 is always 5.
Find whether 8 is a polynomial?
Find whether \frac{(x-2)(x-4)}{x} is a polynomial?
A binomial may have degree 5
Zero of a polynomial is always 0.
A polynomial cannot have more than one zero.
Which of the following expressions are polynomials? Justify your answer:
Factorise 4x^2+20x+25.
Find the factorised form of x^{2}+4 y^{2}+9 z^{2}-4 x y-12 y z+6 x.
Find the factorised form of 16a^2+b^2+4c^2-8ab-4bc+16ac.
Find the product of \left(\frac{x}{2}+2y\right)\left(\frac{x^2}{4}-xy+4y^2\right).
If a + b + c = 9 and ab + bc + ca = 26, find a2 + b2 +c2.
Find the expanded form of \left(4-\frac{1}{3 x}\right)^{3}.
Find the expanded form of \left(\frac{1}{x}+\frac{y}{3}\right)^{3}.
Find the expanded form of (3a-2b)^3.
Factorise 8 p^{3}+\frac{12}{5} p^{2}+\frac{6}{25} p+\frac{1}{125}.
Factorise 1-64a^3-12a+48a^2.
Find the product of \left(x^{2}-1\right)\left(x^{4}+x^{2}+1\right).
Factorise a^3-8b^3-64c^3-24abc.
Find the factorisation of 1+64 x^{3}.
Find the product of (2x-y+3z)\left(4x^2+y^2+9z^2+2xy+3yz-6xz\right).
Without actually calculating the cubes, find the value of (0.2)^{3}-(0.3)^{3}+(0.1)^{3}.
Factorise 2 \sqrt{2} a^{3}+8 b^{3}-27 c^{3}+18 \sqrt{2} a b c.
Without finding the cubes, factorise (x- 2y)3 + (2y – 3z)3 + (3z – x)3.
Find the value of x3 -8y3 -36xy-216,when x = 2y + 6.
Without actually calculating the cubes, find the value of \left(\frac{1}{2}\right)^2+\left(\frac{1}{3}\right)^3-\left(\frac{5}{6}\right)^3.
Give possible expression for the length and breadth of the rectangle whose area is given by 4a2 +4a – 3.
Factorise 9 y^{2}-66 y z+121 z^{2}.
Check whether p(?) is a multiple of g(?) or not:
p(?) = ?3 – 5?2 + 4? – 3, g(?) = ? – 2
Using suitable identity, evaluate 103^3.
Classify wether the given polynomial as polynomials in one variable, two variables, three variables:
x^{2}-2 x y+y^{2}+1
x^2+x+1
y^3-5y
xy + yz + zx
Determine the degree of -10.
Determine the degree of 2x-1.
Determine the degree of y^3 (1 – y^4).
Write the coefficient of x² in \frac{\pi}{6}x+x^2-1.
Classify whether 3 as a constant, linear, quadratic and cubic polynomial.
Determine the degree of x^3 – 9x + 3x^5.
For the polynomial \frac{x^3+2x+1}{5}-\frac{7}{2}x^2-x^6, the constant term.
For the polynomial \frac{x^3+2x+1}{5}-\frac{7}{2}x^2-x^6, the coefficient of x6.
For the polynomial \frac{x^3+2x+1}{5}-\frac{7}{2}x^2-x^6, write the degree of the polynomial.
For the polynomial \frac{x^3+2x+1}{5}-\frac{7}{2}x^2-x^6, the coefficient of x3.
Write the coefficient of x2 in (2x – 5) (2×2 – 3x + 1).
Write the coefficient of x2 in (x –1) (3x – 4).
Classify wether 2-x^2+x^3 as a constant, linear, quadratic and cubic polynomial.
Classify whether 3x^3 as a constant, linear, quadratic and cubic polynomial.
Write the coefficient of x2 in 3x – 5.
Classify whether t^2 as a constant, linear, quadratic and cubic polynomial.
Classify whether 5 t-\sqrt{7} as a constant, linear, quadratic and cubic polynomial.
Classify wether 4 – 5y^2 as a constant, linear, quadratic and cubic polynomial.
Classify whether 2 + x **as a constant, linear, quadratic and cubic polynomial.**
If p(?) =?2 – 4? + 3, evaluate: ?(2)− ?(−1) + ?(½).
Find the corresponding example of trinomial of degree 2.
Find the corresponding example of monomial of degree 1.
Find the corresponding example of binomial of degree 20.
Find the value of the polynomial 3?3 – 4?2 + 7? – 5, when x = -3.
Find the value of the polynomial 3?3 – 4?2 + 7? – 5, when x = 3.
Find p(-2) for (?)=10?−4?^2-3.
Find the zeroes of the polynomial in p(x)=x-4.
By actual division, find the quotient when the first polynomial is divided by the second polynomial: x4 + 1; x –1.
Find the zeroes of the polynomial:
p(?)= (? –2)2−(? + 2)2
Find p(0) for (?)=10?−4?^2-3.
Verify whether the following is true or false:
–3 is a zero of y2 + y – 6
– 4/5 is a zero of 4 –5y
0 and 2 are the zeroes of t2 – 2t
Find p(-2) for (?)=(y + 2) (y – 2).
Find p(0) for (?)=(y + 2) (y – 2).
Find p(1) for (?)=(y + 2) (y – 2).
– 1/3 is a zero of 3x + 1
–3 is a zero of x – 3.
Find the zeroes of the polynomial in h(y)=2y.
Find the zeroes of the polynomial in g(x)=3-6x.
Find the zeroes of the polynomial in q(x)=2x-7.
By actual division, find the remainder when the first polynomial is divided by the second polynomial: x4 + 1; x –1.
By Remainder Theorem find the remainder, when p(x) is divided by g(x), where
p(x)=x^3-3x^2+4x+50,g(x)=x-3
p(x)=x^3-6x^2+2x-4,g(x)=1-\frac{3}{2}x
p(x)=4x^3-12x^2+14x-3,g(x)=2x-1
\mathrm{p}(x)=x^3-2x^2-4x-1,g(x)=x+1
Find the value of x3 +y3 -12xy + 64,when x+y = -4.
p(?)= 2?3 – 11?2− 4? + 5, ?(?)= 2? + 1
State whether 2?−3 is a factor of ? + 2?^3 – 9?^2 + 12.
State whether ?+3 is a factor of 69 + 11?−?^2 + ?^3.
State whether ?-2 is a factor of 4?^2 + ?−2.
State whether ?-2 is a factor of 3?^2 + 6?−24.
State wether p-1 is a factor of p^{11} – 1.
State wether p-1 is a factor of p^{10} – 1.
For what value of m is ?3 – 2??2 + 16 divisible by x + 2?
If ? + 2? is a factor of ?5 – 4?2?3 + 2? + 2? + 3, find a.
Find the value of m, so that 2x -1 be a factor of 8x4 +4x3 -16x2 +10x+07.
Find the factorization of 84-2r-2r^2.
If x +1 is a factor of ax3 +x2 -2x+4a-9, then find the value of a.
Find the factorization of 2x^3-3x^2-17x+30.
Factorise 16x^2 + 4y^2 + 9z^2 – 16xy – 12yz + 24xz.
Find the factorization of 6x^2 +7x -3.
Find the factorization of 2x^2 -7x.-15 .
Find the factorization of x^2+9x+18.
Find the factorization of 3x^3 – x^2 – 3x +1.
Find the factorization of x^3 + x^2 – 4x – 4.
Find the factorization of x^3 -6x^2 +11 x-6.
Factorise 9x^2 +4y^2 + 16z^2 +12xy-16yz -24xz.
Factorise 9x^2 -12x+ 3 .
Using suitable identity, evaluate 999^2.
Using suitable identity, evaluate 101\times102.
Factorise \left(2 x+\frac{1}{3}\right)^{2}-\left(x-\frac{1}{2}\right)^{2}.
Factorise 25x^2 + 16y^2 + 4z^2 – 40xy + 16yz – 20xz.
Factorise 9x^2 -12xy + 4.
Find the factorised form of 9 a^{2}+25 b^{2}+c^{2}-30 a b+10 b c-6 a c.
Classify the following polynomials as polynomials in one variable, two variables etc.
(i) x2 + x
(ii) y3 – 5y
(iii) xy + yz + zx
(iv)x2 – 2xy + y2 + 1
Determine the degree of each of the following polynomials:
(i) 2x – 1
(ii) –10
(iii) x3 – 9x + 3×5
(iv) y3 (1 – y4)
For the polynomial
\frac{x^{3}+2 x+1}{5}-\frac{7}{2} x^{2}-x^{6}
, write
(i) the degree of thepolynomial
(ii) the coefficient ofx3
(iii) the coefficient ofx6 the constant term
Write the coefficient of x2 in each of the following:
(i) (π/6)x + x2 – 1
(ii) 3x – 5
(iii) (x –1) (3x – 4)
(iv) (2x – 5) (2×2 – 3x + 1)
Classify the following as a constant, linear, quadratic and cubic polynomials:
(i) 2 – x2 + x3
(ii) 3×3
(iii) 5t – √7
(iv) 4 – 5y2
(v) 3
(vi) 2 + x
(vii) y3 – y
(viii) 1 + x + x2
(ix) t2
(x) √2x – 1
Give an example of a polynomial, which is:
(i) monomial of degree1
(ii) binomial of degree20
(iii) trinomial of degree 2
Find the value of the polynomial 3?3 – 4?2 + 7? – 5,
when x = 3 and also when x = –3.
If p(?) =?2 – 4? + 3,
evaluate: ?(2)− ?(−1) + ?(½).
Find p(0), p(1),?(−2) for the following polynomials:
(i) (?)=10?−4?2 –3
(ii) (?)=(y + 2) (y – 2)
Verify whether the following are true or false:
(i) –3 is a zero of x – 3
(ii) – 1/3 is a zero of 3x + 1
(iii) – 4/5 is a zero of 4 –5y
(iv) 0 and 2 are the zeroes of t2 – 2t
(v)–3 is a zero of y2 + y – 6
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 Solution:
By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial:
x4 + 1; x –1
By Remainder Theorem find the remainder, when p(x) is divided by g(x), where (i) p(?) = ?3 – 2?2 – 4? – 1, g(?) = ? + 1
(ii) p(?) = ?3 – 3?2 + 4? + 50, g(?) = ? – 3 (iii) p(?) = 4?3 – 12?2 + 14? – 3, g(?) = 2? – 1 (iv) p(?) = ?3 – 6?2 + 2? – 4, g(?) = 1 – 3/2 ?
(i) p(?) = ?3 – 5?2 + 4? – 3, g(?) = ? – 2
(ii) p(?)= 2?3 – 11?2− 4? + 5, ?(?)= 2? + 1
Show that:
(i) ? + 3 is a factor of
69 + 11?−?2 + ?3.
(ii) 2?−3 is a factor of
? + 2?3 – 9?2 + 12
Determine which of the following polynomials has x – 2 a factor:
(i) 3?2 + 6?−24.
(ii) 4?2 + ?−2.
Show that p – 1 is a factor of p10 – 1 and also of p11 – 1.
If ? + 2? is a factor of ?5 – 4?2?3 + 2?+ 2? + 3, find a.
State whether the given statement is true or false.
(a +b +c)3 -a3 -b3 – c3 =3(a +b)(b +c)(c +a).
If bothx– 2 andx– ½ are factors of px2+ 5x+r, state whether p=r.
Multiply x2 + 4y2 + z2 + 2xy + xz – 2yz by (-z + x-2y).
If a, b, c are all non-zero and a + b + c = 0, state wether \frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}=3.
If a+b+c= 5 and ab+bc+ca =10, then state whether that a3 +b3 +c3 – 3abc = -25.
Simplify (2x- 5y)3 – (2x+ 5y)3.
Without actual division, check if 2x4– 5x3+ 2x2– x + 2 is divisible by x2– 3x + 2.
If the polynomials az3 + 4z2 + 3z – 4 and z3 – 4z + a leave the same remainder when divided by z – 3, find the value of a.
The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by x + 1 leaves the remainder 19. Find the remainder when p(x) is divided by x + 2.
If the polynomials az3 + 4z2 + 3z – 4 and z3 – 4z + leave the same remainder when divided by z – 3, find the value of a.
The polynomial p(x) = x4 – 2×3 + 3×2 – ax + 3a – 7 when divided by x + 1 leaves the remainder 19. Find the values of a. Also find the remainder when p(x) is divided by x + 2.
If both x – 2 and x – ½ are factors of px2 + 5x + r, show that p = r.
Without actual division, prove that 2×4 – 5×3 + 2×2 – x + 2 is divisible by x2 – 3x + 2.
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