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Write whether the following statements are true of false : In each case justify your answer .
(i) \(\dfrac{1}{\sqrt5}x^{\frac{1}{2}}+1 \) is a polynomial
(ii) \(\dfrac{6\sqrt x+x^{\frac{3}{2}}}{\sqrt x} \) is a polynomial, \(x\ne0 \).
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(i) False, because its power is not on integer
(ii)
is a polynomial as at power is an integer
Find the zeroes of the polynomial \(p(x) = x^25x+6. \)
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The zeroes are 2 and 3
For the polynomial \(\dfrac{x^3+2x+1}{5}\dfrac{7}{2}x^2x^5 \), write
(i) The degree of the polynomial
(ii) The coefficient of \(x^3 \)
(iii) The coefficient of \(x^6 \)
(iv) The constant term
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(i) The degree of the polynomial 5
(ii) The coefficient of is
(iii) The coefficient of is
(iv) The constant term is
Give an example of a polynomial which is :
(i) Monomial of degree 1
(ii) Binomial of degree 20
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(i) The example of monomial of degree is
(ii)
Show that \(x+3\) is a factor of \(69 + 11x – x^2 + x^3\).
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Verify that 1 is not a zero of the polynomial \(4y^4 – 3y^3 + 2y^2 – 5y + 1 \).
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Let
So 1 is not a zero of the polynomial
Evalute using suitable identity \((999)^3 \).
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Factorise : \(x^2+\dfrac{x}{4}\dfrac{1}{8} \).
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Factorise : \(27p^3\dfrac{1}{216}\dfrac{9}{2}p^2+\dfrac{1}{4}p \).
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If \(x^2+\dfrac{1}{x^2}=38 \). then find the value of \(\left(x\dfrac{1}{x}\right) \).
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Find the product of \(\left(x\dfrac{1}{x}\right),\left(x+\dfrac{1}{x}\right),\left(x^2+\dfrac{1}{x^2}\right) \) and \(\left(x^4+\dfrac{1}{x^4}\right) \).
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Check whether the polynomial \(p(x) = 3s^3+s^220s+12 \) is a multiple of \(3s2 \).
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Check whether \(p(x)\) is a multiple of \(g(x)\) or not, where \( p(x)=x^3x+1,\ g(x) = 2 – 3x.\)
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So is not a multiple of
Check whether \(g(x)\) is a factor of \(p(x) \) or not, where: \(p(x) = 8x^3 – 6x^2 – 4x + 3,g(x) =\dfrac{x}{3}\dfrac{1}{4} \)
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So is not a factor of
What must be added to \(x^33x^212x+19 \) so that the result is exactly divisible by \(x^2+x6 \)?
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So must be added
Using suitable identity, evaluate the following \(\mathrm{(i)\ 103^3\ (ii)\ 101 \times 102\ (C)\ 999^2}\)
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Find the following product : \((2xy+3z)\ (4x^2+y^2+9z^2+2xy+3yz6xz) \).
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Find the value of \(x^3+y^312xy+64 \) when \(x+y=4 \).
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If \(2a=3+2b \), prove that \(8a^38b^336ab=27 \)
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Cubing both the sides
The polynomials \(kx^3+3x^28 \) and \(3x^35x+k \) are divided by \(x+2 \). If the remainder in each case is the same, find the value of \(k \).
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A/Q
Factorise : \(4(x^2+1)^2+13(x^2+1)12 \)
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Factorise : \(x^2+\dfrac{1}{x^2}+22x\dfrac{2}{x} \).
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Simplify : \(\left(x\dfrac{2}{3}y\right)^3\left(x+\dfrac{2}{3}y\right)^3 \).
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Using suitable identity evaluate : \((42)^3 – (18)^3 – (24)^3 \)
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Let
Factorise \(9x^2+y^2+z^26xy+2yz6zx \). Hence find the value if \(x=1,y=2 \) and \(z=1 \).
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If \(a + b + c = 6, \) then find the value of \((2a)^3 +(2b)^3 +(2c)^33(2a)(2b)(2c). \)
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Let
If \(a^2+b^2+c^2=250 \) and \(ab+bc+ca=3 \). Find \( a+b+c\).
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If \(ax^3+bx^2+x6 \) has \((x+2) \) as a factor and leaves a remainder 4 when divided by \(x2 \), find the value of \(a \) and \(b \).
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….(1)
….(2)
Solve equation (1) and (2)
and
If \(p=4q \), prove that \(p^3+q^3+12pq=64 \)
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What are the possible expressions for the dimensions of the cuboids whose volume is given below ?
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Without finding the cubes, find the value of : \(\left(\dfrac{1}{4}\right)^3+\left(\dfrac{1}{3}\right)^3\left(\dfrac{7}{12}\right)^3 \).
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Factorise : \(2\sqrt2a^3+8b^327c^3+18\sqrt2abc \).
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( Using the identity
Factorise : \(x^6y^6 \).
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If both \((x2) \) and \(\left(x\dfrac{1}{2}\right) \) are factors of \(px^2+5x+r \), show that \(p=r \).
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….(1)
… (2)
Solving equation (1) and (2)
So,
Find the value of \(a \) if \((x+a) \) is a factor of \(x^4a^2\ x^2+3xa \)
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Factorise by spliting the middle term : \(9(x2y)^24(x2y)13 \).
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Let
Without actual division prove that \((x2) \) is a factor of the polynomial \(3x^313x^2+8x+12 \) Also, factorise it completely.
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If \(a,b,c \) are all nonzero and \(a+b+c=0 \), prove that \(\dfrac{a^2}{bc}+\dfrac{b^2}{bc}+\dfrac{c^2}{ab}=3 \)
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Multiplying by
Prove that \((a+b+c)^3a^3b^3c^3=3\ (a+b)\ (b+c)\ (c+a) \).
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If \(a + b + c = 5 \) and \(ab + bc + ca = 10 \), then prove that \(a^3+b^3+c^33abc=25 \).
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Simplify : \(\dfrac{(a^2b^2)^3+(b^2c^2)^3+(c^2a^2)^3}{(ab)^3+(bc)^3+(ca)^3} \)
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Again Let
Prove that : \(2x^3+2y^3+2z^36xyz=(x+y+z)\ [(xy)^2+(yz)^2+(zx)^2] \). Hence evaluate \(2(7)^3 + 2(9)^3 + 2(13)^3 – 6(7)\ (9) (13) . \)
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RHS
Using Identity
Using factor theorem show that \(x^2+5x+6 \) is factor of \(x^4+5x^3+9x^2+15x+18 \).
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zeroes are
So is a factor of
Prove that \( (x+y+z)\times[(xy)^2+(yz)^2]=2(x^3+y^3+z^33xyz)\)
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A/Q
The polynomial \(p(x) =ax^3 + 4x^2 + 3x – 4 \) and \(q(x) = x^3 – 4x + a \) leave the same remainder when divided by \(x3 \). Find the remainder when \(p(x) \) is divided by \((x2) \).
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Simplify by factrorisation method : \(\dfrac{62\sqrt2xx^2}{2x^2} \)
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The polynomials \(x^3+2x^25ax8 \) and \(x^3+ax^212x6 \) when divided by \((x2) \) and \((x3) \) leave reminders \(p \) and \(q \) respectively. If \(qp=10 \), find the value of \(a \)
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Prove that \((x+y)^3(xy)^36y(x^2y^2)=8y^3 \).
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Find the value of \((xa)^3+(xb)^3+(xc)^33(xa)(xb)(xc) \), if \(a + b + c = 3x. \)
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Simplify by factorisation method : \(\dfrac{92\sqrt3xx^2}{3x^2} \)
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Adding equation (1) and (2)
If \(p(x)=x^3ax^2+bx+3 \) leaves a remainder \(19 \) when divided by \((x+2) \) and a remainder \( 17\) when divided by \((x2) \), prove that \(a+b=6 \).
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Side
when side
Volume
Volume
The volume of a cube is given by the polynomial \(p(x) = x^3 – 6x^2 + 12x – 8 \). Find the possible expression for the sides of the cube. Verify the truth of your answer when the length of cube is 3 cm.
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Side
when side
Volume
Volume
Using factor theorem, factorise the polynomial: \(x^4 + 3x^3 + 2x^2 – 3x – 3. \)
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Using factor theorem, factorise the polynomial. \(x^4+2x^37x^28x+12 \).
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Without actual division, show that the polynomail \(2x^45x^3+2x^2x+2 \) is exactly divisible by \(x^23x+2 \)
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If \(x \) and \(y \) be two positive real numbers such that \(8x^3+27y^3=730 \) and \(2x^2y+3xy^2=15 \), then evaluate \(2x+3y \).
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Without actual division prove that \(x^4+ 2x^3 2x^2+ 2x – 3 \) is exactly divisible by \(x^2+2x3. \)
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Multiply \(9x^2+25y^2+15xy+12x20y+16 \) by \(3x5y4 \) using suitable identify.
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Without actual division show that \(x^4+2x^32x^2+2x3 \) is exactly divisible by \(x^2 + 2x – 3. \)
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Factorise : \(27a^3+\dfrac{1}{64b^3}+\dfrac{27a^2}{4b}+\dfrac{9a}{16b^2} \).
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Find the value of \(a \) and \(b \) so that \((x+1) \) and \((x2) \) are factors of \((x^3+ax^2+2x+b) \).
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…(1)
… (2)
Subtracting equation (1) from (2) we get
Without actual division, prove that \((2x^4 – 6x^3 + 3x^2 + 3x – 2) \) is exactly divisible by \((x^2 – 3x + 2) \)
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Find the value of \(a \) if \((xa) \) is a factor of \(x^5a^2x^3+2x+a+3 \), hence factorise \(x^22ax3 \)
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The polynomial \(ax^3+3x^23 \) and \(2x^35x+a \) when divided by \(x4 \) leave the same remainder in each case. Find the value of \(a \).
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A/Q
If \(x+\dfrac{1}{x}=5 \), then evaluate \(x^6+\dfrac{1}{x^6} \).
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If then
Without actual division, prove that \(2x^48x^3+3x^2+12x9 \) is exactly divisible by \(x^24x+3 \).
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If \(f(x) = x^4 – 2x^3 + 3x^2 – ax + b \) is divided by \((x1) \) and \((x+1) \), it leaves the remainder 5 and 19 respectively. Find \(a \) and \(b \).
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…(1)
….(2)
Adding equation (1) and (2)