List of top Mathematics Questions on Polynomials

The graph of \(y = f(x)\) is shown in the figure for some polynomials \(f(x)\). The number of zeroes for the polynomials is
Assertion (A) : The polynomial $p(y) = y^2 + 4y + 3$ has two zeroes.
Reason (R) : A quadratic polynomial can have at most two zeroes.
If \(\alpha\) is a root of the equation \[ x^4-4x^3+16x-16=0 \] of multiplicity \(m\), then \[ m^2-4m+2= \]
If \(\alpha,\beta,\gamma,\delta\) are the roots of the equation \[ x^4+x^3-x-1=0, \] then \[ \alpha^3+\beta^3+\gamma^3+\delta^3= \]
Case Study - 2 : During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial p(x) = -x\(^2\) + 2x + 8, where x is the length (in feet) on stage level. Based on the figure, answer the following questions : (i) Determine the height of the arch. (ii) (a) Find zeroes of the polynomial p(x). Which points on the graph represent the zeroes? OR (ii) (b) Find the span of the arch on the stage floor. (iii) Write the coordinates of the point of intersection of the above curve with the y-axis.
A polynomial \(p(x)\), which has sum of its zeroes equal to their product, is :
During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial \(p(x) = -x^2 + 2x + 8\), where x is the length (in feet) on stage level.
Observe the figure and answer the following questions :
(i) Determine the height of the arch.
(ii)(a) Find zeroes of the polynomial p(x). Which points on the graph represent the zeroes?
OR
(ii)(b) Find the span of the arch on the stage floor.
(iii) Write the coordinates of the point of intersection of the above curve with the y-axis.
Case Study - 1: During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial $p(x) = -x^2 + 2x + 8$, where x is the length (in feet) on stage level. Based on the figure given, answer the following questions :

(i) Determine the height of the arch.
How many zeroes does \(p(x) = (x - 2)(x + 3)\) have ?
Find the value of p, for which one zero of the quadratic polynomial \(px^2 - 14x + 8\) is 6 times the other.
The graph of y = f(x) is given. The number of distinct zeroes of y = f(x) is :
The graph of y = f(x) is given. The number of distinct zeroes of y = f(x) is :
The graph of $y = f(x)$ is given. The number of zeroes of $f(x)$ is :
Assertion (A) : The polynomial $p(y) = y^2 + 4y + 3$ has two zeroes.
Reason (R) : A quadratic polynomial can have at most two zeroes.
The graph of $y = f(x)$ is given. The number of zeroes of $f(x)$ is :
If \(\alpha\) and \(\beta\) are two zeroes of the quadratic polynomial \(p(x) = x^2 - 11x + 30\), then \(\frac{1}{\alpha} + \frac{1}{\beta}\) is equal to :
The graph of \(y = f(x)\) is given. The number of zeroes of \(f(x)\) is :
If \(5\) is the remainder when \[ 2x^5+kx^4+5x^3-3x^2+2x-1 \] is divided by \[ x^2+x+1, \] then the quotient is:
2 is a zero of the polynomial function \[ f(x)=x^4+kx^3+22x^2-6x-20. \] If \(-2,\alpha,\beta\) are the roots of the equation \[ x^3+3x^2+2kx-40=0 \] and \(\alpha<\beta\), then \(2\alpha+3\beta=\)
If \( x^5 + ax^4 + bx^3 + cx^2 + 5x + d = 0 \) is an odd order reciprocal equation of second type and \( \frac{1+\sqrt{3}i}{2} \) is a root, then \( b-c = \)
If \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 - 3x + 1 = 0 \), then \( \frac{\alpha}{1-\alpha} + \frac{\beta}{1-\beta} + \frac{\gamma}{1-\gamma} = \)
If \( x^{5}+ax^{4}+b death^{3}+cx^{2}+5x+e=0 \) is a reciprocal equation of second kind such that \( a+b=4 \) then the number of complex roots of this equation is
The equation formed with the roots obtained by diminishing the roots of the equation \( x^{4}+3x^{3}-7x^{2}+4x+1=0 \) by 'h', does not contain the \( x^{2} \) term. If the possible values of such 'h' are \( h_{1}<0 \) and \( h_{2}>0, \) then which one of the following is true?
If $\alpha, \beta, \gamma$ are the zeroes of $4x^3 + 8x^2 - 6x - 2$, then the value of $\alpha\beta + \beta\gamma + \gamma\alpha$ is _____.
If \( \alpha \) and \( \beta \) are two zeroes of a polynomial \( f(x) = px^2 - 2x + 3p \) and \( \alpha + \beta = \alpha\beta \), then value of p is :