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If \(\alpha \) and \(\beta,\gamma \) are zeroes of the polynomial \(6x^3+3x^25x+1 \), then find the value of \(\alpha^{1}+\beta^{1}+\gamma^{1} \).
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If the zeroes of the polynomial \(x^33x^2+x+1 \) are \(ab \) and \(a+b \), find the value of \(a \) and \(b \).
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sum of zeroes
On dividing \( x^33x^2+x+2\) by a polynomial \(g(x) \), the quotient and remainder were \(x2 \) and \(2x+4 \) respectively. Find \(g(x) \)
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Equation
If \(\alpha,\beta \) are zeroes of the polynomial \(x^22x8 \), then form a quadratic polynomial whose zeroes are \( 2\alpha\) and \(2\beta \).
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Product
Obtain all zeroes of \(f(x)=x^43x^3x^2+9x6 \) if two of its zeroes are \((\sqrt3) \) and \(\sqrt3 \).
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The zeroes are
Check whether the polynomial \(g(x)=x^33x+1 \) is the factor of polynomial \(px(x)=x^54x^3x^2+3x+1 \)
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No it is not a factor
If \(\alpha,\beta \) are the zeroes of the polynomial \(25p^215p+2 \), find a quadratic polynomial whose zeroes are \(\dfrac{1}{2\alpha} \) and \(\dfrac{1}{2\beta} \).
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Divide \(3x^33x+5 \) by \(x1x^2 \) and verify the division algorithm
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Verify
If \(\alpha,\beta \) are the zeroes of the polynomial \(21y^2y1 \), find a quadratic polynomial whose zeroes are \( 2\alpha\) and \(2\beta \).
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Find the zeroes of \(4\sqrt5x^2+17x+3\sqrt5 \) and verify the relation between the zeroes and coefficients of the polynomial.
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If the polynomial \(6x^4+8x^3+17x^2+21x+7 \) is divided by another polynomial \(3x^2+4x+1 \), the remainder comes out to be \((ax+b) \), find \(a \) and \(b \).
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A/Q
Find all the zeroes of the polynomial \(2x^3+x^26x3 \), if two of its zeroes are \(\sqrt3 \) and \(\sqrt3 \)
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The zeroes are
If \(\alpha \) and \(\beta \) are the zeroes of the quadratic polynomial \(p(s)=3s^26s+4 \), find the value of \(\dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}+2\left(\dfrac{1}{\alpha}+\dfrac{1}{\beta}\right)+3\alpha\beta \).
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If \(\alpha \) and \(\beta \) are the zeroes of the quadratic polynomial \(f(x)=x^2px+q \), prove that \(\dfrac{\alpha^2}{\beta^2}+\dfrac{\beta^2}{\alpha^2}=\dfrac{p^4}{q^2}\dfrac{4p^2}{q}+2 \).
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(Proved)
If \(\alpha \) and \(\beta \) are the zeroes of the quadratic polynomial \(f(x)=x^2+px+q \), form a polynomial whose zeroes are \((\alpha+\beta)^2 \) and \((\alpha\beta)^2 \)
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If \(\alpha \) and \(\beta \) are the zeroes of the polynomial \(f(x)=x^22x+3 \), find a polynomial whose zeroes are \(\alpha+2 \) and \(\alpha+\beta \).
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Product of roots
– (sum of roots) + Product of root
Obtain all the zeroes of the polynomial \(f(x) = 2x^4+x^314x19x6 \), if two of its zeroes are \( 2\) and \(1 \).
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What must be added to \(6x^5+5x^4+11x^33x^2+x+5 \) so that it may be exactly divisible by \( 3x^22x+4\)?
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So must be added
What must be subtracted from the polynomial \(f(x)= x^4+2x^313x^212x+21 \) so that the resulting polynomial is exactly divisible by \(g(x) = x^24x+3 \)?
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So must be subtracted
Find the other zeroes of the polynomial \(2x^43x^33x^2+6x2 \), if \(\sqrt2 \) and \( \sqrt2\) are the zeroes of the given polynomial.
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If the remainder on division of \(x^3+2x^2+kx+3 \) by \(x3 \) is \(21 \), find the quotient and the value of \( k\). Hence, find the zeroes of the cubic polynomial \(x^3+2x^2+kx18 \).
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The zeroes are
If two zeroes of \( p(x) = x^46x^326x^2+138x–35 \) are \(2\pm\sqrt3 \), find the other zereos.
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other two zeroes are 5 and 7
Find other zeroes of the polynomial \( x^4+x^39x^23x+18 \), if it is given that two of its zeroes are \(\sqrt3 \) and \(\sqrt3 \).
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are zeroes.
Divide \(30x^4+11x^382x^212x+48 \) by \( (3x^2+2x4)\) and verify the result by division algorithm.
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Verify
Find all the zeros of the polynomial \(2x^410x^3+5x^2+15x12 \), if it is given that two of its zeroes are \(\sqrt{\dfrac{3}{2}} \) and \(\sqrt{\dfrac{3}{2}} \).
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Find all the zeroes of the polynomial \(2x^43x^25x^2+9x3 \), it being given that two of its zeroes are \(\sqrt3 \) and \(\sqrt3 \)
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Zeroes of polynomial are
What must be added to the polynomial \(p(x)=5x^4+6x^313x^244x+7 \) so that the resulting polynomial is exactly divisible by the polynomial \(Q(x)=x^2+4x+3 \) and the degree of the polynomial to be added must be less than degree of the polynomial \(Q(x) \).
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So, should be added
If the polynomial \(6x^4+8x^35x^2+ax+b \) is exactly divisible by the polynomial \(2x^25 \), then find the value of \(a \) and \( b\).
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