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The graph of \(y=f(x) \) cuts the xaxis at \((1,0) \) and \(\left(\dfrac{3}{2},0\right) \). Find all the zeroes of \(f(x) \)
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at and
The zeroes of the polynomial are and
Show that \(1,1 \) and \(3 \) are the zeroes of the polynomial \(x^33x^2x+3 \).
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Write the polynomial, the product and sum of whose zeroes are \(\dfrac{9}{2} \) and \(\dfrac{3}{2} \) respectively
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– (Sum of zeroes) + product of zeroes
If \((x+a) \) is a factor of \(2x^2+2ax+5x+10 \), find \( a\).
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If \(\alpha \) and \(\beta \) are the zeroes of the quadratic polynomial \(f(t)=t^24t+3 \), find the value of \(\alpha^4\beta^4+\alpha^3\beta^4 \)
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Find a quadratic polynomial, the sum and product of whose zeroes are \(\sqrt2 \) and \(\dfrac{1}{3} \) respectively.
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Divide \(6x^3+13x^2+x2 \) by \(2x+1 \), and find the quotient and remainder.
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Divide \(x^43x^2+4x+5 \) by \(x^2x+1 \), find quotient and remainder.
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\(\alpha,\beta \) are the roots of the quadratic polynomial \(p(x)=x^2(k+6)x+2\ (2k1) \). Find the value of \(k \), if \(\alpha+\beta=\dfrac{1}{2}\alpha\beta \)
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A/Q
\(\alpha,\beta \) are the roots of the quadratic polynomial \(p(x)=x^2(k+6)x+2\ (2k1) \). Find the value of \(k \), if \(\alpha+\beta=\dfrac{1}{2}\alpha\beta \)
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A/Q
What must be added to polynomial \(f(x)=x^4+2x^32x^2+x1 \) so that the resulting polynomial is exactly divisible by \(x^2+2x3 \).
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must be added
What must be added to polynomial \(f(x)=x^4+2x^32x^2+x1 \) so that the resulting polynomial is exactly divisible by \(x^2+2x3 \).
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must be added
Find a quadratic polynomial whose zeroes are 2 and –6. Ve rify the relation between the coefficients and zeroes of the polynomial.
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Polynomial
Sum of zeroes
Product of zeroes
Find a quadratic polynomial whose zeroes are 2 and –6. Ve rify the relation between the coefficients and zeroes of the polynomial.
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Polynomial
Sum of zeroes
Product of zeroes
Can (x + 3) be the remainder on the division of a polynomial p(x) by (2x– 5)? Justify your answer.
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cannot be the remainder on the division of a polynomial by because the degree of remainder should be less than the degree of divisor. Here the degree of remainder is equal to the degree of divisor.
Can (x + 3) be the remainder on the division of a polynomial p(x) by (2x– 5)? Justify your answer.
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cannot be the remainder on the division of a polynomial by because the degree of remainder should be less than the degree of divisor. Here the degree of remainder is equal to the degree of divisor.
Find the zeroes of the polynomial \(2x^27x+3 \) and hence find the sum of product of its zeroes.
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Sum of zeroes
Product
Find the zeroes of the polynomial \(2x^27x+3 \) and hence find the sum of product of its zeroes.
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Sum of zeroes
Product
Check whether \(x^2+3x+1 \) is a factor of \(3x^4+5x^37x^2+2x+2 \).
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Yes it is a factor
Check whether \(x^2+3x+1 \) is a factor of \(3x^4+5x^37x^2+2x+2 \).
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Yes it is a factor
Find the zeroes of the quadratic polynomial \(x^2+7x+12 \) and verify the relationship between the zeroes and its coefïcients.
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Sum , Product
Sum of zeroes , Product
Find the zeroes of the quadratic polynomial \(x^2+7x+12 \) and verify the relationship between the zeroes and its coefïcients.
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Sum , Product
Sum of zeroes , Product
Divide \((2x^2+x20) \) by \((x+3) \) and verify division algorithm.
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Verify
Divide \((2x^2+x20) \) by \((x+3) \) and verify division algorithm.
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Verify
If \(\alpha \) and \(\beta \) are the zeroes of \(x^2+7x+12 \), then find the value of \(\dfrac{1}{\alpha}+\dfrac{1}{\beta}2\alpha\beta \)
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If \(\alpha \) and \(\beta \) are the zeroes of \(x^2+7x+12 \), then find the value of \(\dfrac{1}{\alpha}+\dfrac{1}{\beta}2\alpha\beta \)
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If \(\alpha \) and \(\beta \) are the zeroes of the polynomial \(2y^2+7y+5 \), write the value of \(\alpha+\beta+\alpha\beta \).
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If \(\alpha \) and \(\beta \) are the zeroes of the polynomial \(2y^2+7y+5 \), write the value of \(\alpha+\beta+\alpha\beta \).
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If the product of zeroes of the polynomial \(ax^26x6 \) is \(4 \), find the value of \( a\).
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If the product of zeroes of the polynomial \(ax^26x6 \) is \(4 \), find the value of \( a\).
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Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.
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quadratic polynomial – (sum of zeroes) + Product of zeroes
zeroes are 2 and 6
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.
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quadratic polynomial – (sum of zeroes) + Product of zeroes
zeroes are 2 and 6
If one zero of the polynomial \((a^2+9)x^2+13x+6a \) is reciprocal of the other, find the value of \( a\).
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Let the zeroes be and
If one zero of the polynomial \((a^2+9)x^2+13x+6a \) is reciprocal of the other, find the value of \( a\).
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Let the zeroes be and
Find the zeroes of the quadratic polynomial \(f(x)=abx^2+(b^2ac)xbc \) and verify the relationship between the zeroes and its coefficients.
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sum
Product
Find the zeroes of the quadratic polynomial \(f(x)=abx^2+(b^2ac)xbc \) and verify the relationship between the zeroes and its coefficients.
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sum
Product
Find the zeroes of the quadratic polynomial \(p(x)=x^2(\sqrt3+1)x+\sqrt3 \) and verify the relationship between the zeroes and its coefficients.
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sum
product
Find the zeroes of the quadratic polynomial \(p(x)=x^2(\sqrt3+1)x+\sqrt3 \) and verify the relationship between the zeroes and its coefficients.
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sum
product
Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time and product of its zeroes as 3, –1 and –3 respectively.
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Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time and product of its zeroes as 3, –1 and –3 respectively.
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If \(\alpha \) and \(\beta \) are zeroes of the quadratic polynomial. \(f(x)=x^21 \), find a quadratic polynomial whose zeroes and \(\dfrac{2\alpha}{\beta} \) and \(\dfrac{2\beta}{\alpha} \)
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1 and 1 are zeroes
Required polynomial
If \(\alpha \) and \(\beta \) are zeroes of the quadratic polynomial. \(f(x)=x^21 \), find a quadratic polynomial whose zeroes and \(\dfrac{2\alpha}{\beta} \) and \(\dfrac{2\beta}{\alpha} \)
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1 and 1 are zeroes
Required polynomial
If the square of the difference of the zeroes of the quadratic polynomial \(f(x)=x^2+px+45 \) is equal to 144, find the value of \(p \).
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If the square of the difference of the zeroes of the quadratic polynomial \(f(x)=x^2+px+45 \) is equal to 144, find the value of \(p \).
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If the sum of the zeroes of the quadratic polynomial \(f(t)=kt^2+2t+3k \) is equal to their product, find the value of \(k \).
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sum of zeroes
Product of zeroes
A/Q
If the sum of the zeroes of the quadratic polynomial \(f(t)=kt^2+2t+3k \) is equal to their product, find the value of \(k \).
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sum of zeroes
Product of zeroes
A/Q
Find the zeroes of the quadratic polynomial \(x^2+\dfrac{7}{2}x+\dfrac{3}{4} \), and verify relationship between the zeroes and the coefficients.
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root are and
sum of roots
Product
Find the zeroes of the quadratic polynomial \(x^2+\dfrac{7}{2}x+\dfrac{3}{4} \), and verify relationship between the zeroes and the coefficients.
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root are and
sum of roots
Product
If \(\alpha \) and \(\beta \) are zeroes of the quadratic polynomial \(x^26x+a \); find the value of \(a \) if \(3\alpha+2\beta=20 \).
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….(1)
…(2)
on solving we get
If \(\alpha \) and \(\beta \) are zeroes of the quadratic polynomial \(x^26x+a \); find the value of \(a \) if \(3\alpha+2\beta=20 \).
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….(1)
…(2)
on solving we get
Divide \((6+19x+x^26x^2) \) by \((2+5x3x^2) \) and verify the division algorithm
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Verify
Divide \((6+19x+x^26x^2) \) by \((2+5x3x^2) \) and verify the division algorithm
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Verify