Practice Questions

Define a quadratic polynomial and state its general form.

The graph of $y = p(x)$ is a curve that intersects the x-axis at three distinct points. Identify the number of zeroes of $p(x)$ and explain your reasoning.

Calculate the value of the cubic polynomial $p(t) = t^3 - 3t^2 + 5t - 6$ at $t = -2$.

Examine the polynomial $4u^2 + 8u$ and determine its zeroes.

What is the maximum number of zeroes a cubic polynomial can have?

Identify the degree of the polynomial $p(u) = 7u^6 - \frac{3}{2}u^4 + 4u^2 + u - 8$.

A classmate states that if the graph of a polynomial is a parabola that touches the x-axis at only one point, it must be a linear polynomial. Critique this statement.

Analyze the expressions $3x^2 - 5\sqrt{x} + 7$ and $x^2 - 9$. Explain why one is not a polynomial while the other is a quadratic polynomial.

Recall the formula for the sum of zeroes of the quadratic polynomial $ax^2 + bx + c$.

Create a quadratic polynomial whose graph is a parabola opening upwards and has exactly one zero at $x=5$.

Solve for the zeroes of the quadratic polynomial $p(x) = 6x^2 - x - 2$ and demonstrate the relationship between the zeroes and the coefficients.

Compare the number of zeroes for two polynomials, $p(x)$ and $q(x)$. The graph of $y=p(x)$ is a straight line parallel to the x-axis that passes through $(0, 3)$. The graph of $y=q(x)$ is a cubic curve that intersects the x-axis at $x=-1, x=0,$ and $x=1$. Analyze each graph to justify your answer.

Analyze the cubic polynomial $p(x) = x^3 - 6x^2 + 11x - 6$. Given that $1, 2,$ and $3$ are its zeroes, demonstrate that the relationships between the zeroes and the coefficients hold true.

Evaluate the claim that a polynomial of degree $n$ must intersect the x-axis at exactly $n$ points. Provide a counterexample using a cubic polynomial and justify your choice.

Create a cubic polynomial $p(x)$ for which the sum of its zeroes is 6, the sum of the products of its zeroes taken two at a time is 11, and the product of its zeroes is 6.

Name the shape of the graph corresponding to the equation $y = ax^2 + bx + c$, where $a \neq 0$.

Explain the geometrical meaning of the zeroes of a polynomial $p(x)$.

Recall the zero of a linear polynomial $p(x) = ax+b$ and explain how it is derived.

List the relationships between the zeroes and coefficients of a cubic polynomial $ax^3+bx^2+cx+d$.

Describe the three possible cases for the number of zeroes of a quadratic polynomial based on the intersection of its graph with the x-axis.

Find a quadratic polynomial if the sum and product of its zeroes are $-3$ and $2$, respectively.

Analyze the graph of a polynomial $y = p(x)$ which is a parabola opening upwards and its vertex is at $(2, -1)$. How many zeroes does the polynomial have?

Solve for the zeroes of the quadratic polynomial $p(t) = t^2 - 17$ and verify the relationship with its coefficients.

Apply the concept of zeroes to find a quadratic polynomial where the sum of its zeroes is $\sqrt{2}$ and the product of its zeroes is $-3/2$.

Evaluate the relationship between the zeroes of the polynomial $p(x) = 4x^2 - 4x + 1$ and its coefficients. Then, design a new quadratic polynomial whose zeroes are $2\alpha$ and $2\beta$, where $\alpha$ and $\beta$ are the zeroes of $p(x)$.

Propose a method to determine the number of real zeroes of a quadratic polynomial $ax^2 + bx + c$ without finding the zeroes or drawing its graph. Justify your proposed method.

Describe the geometrical representation of a linear and a quadratic polynomial, explaining how their graphs relate to their number of zeroes.

Find the value of the polynomial $p(x) = 5x^3 - 4x^2 + x - \sqrt{2}$ at $x=2$.

Apply the relationship between zeroes and coefficients to find a quadratic polynomial whose zeroes are $-5$ and $4$.

If the sum of the zeroes of the quadratic polynomial $f(t) = kt^2 + 2t + 3k$ is equal to their product, calculate the value of $k$. (Assume $k \neq 0$)

Propose a condition on the coefficients of a quadratic polynomial $ax^2 + bx + c$ that would ensure its two zeroes are equal in magnitude but opposite in sign.

Analyze the polynomial $p(x) = x^3 - x$ by factoring it, and determine the number of times its graph intersects the x-axis.

Formulate a quadratic polynomial $p(x)$ where one zero is the negative reciprocal of the other. Justify the relationship between its coefficients $a$ and $c$.

If the sum of the zeroes of the quadratic polynomial $p(x) = kx^2 + 2x + 3k$ is equal to their product, justify the value of $k$.

Summarize the relationship between the zeroes and coefficients for a quadratic polynomial $ax^2+bx+c$. Also, explain how these relationships are derived.

The graph of a quadratic polynomial $y = p(x)$ has its vertex at $(3, -2)$ and passes through the point $(1, 6)$. Formulate the quadratic polynomial $p(x)$ and evaluate its zeroes.

A polynomial is given by $p(x) = (x-a)(x-b) + (x-b)(x-c) + (x-c)(x-a)$. Justify that this polynomial must be a quadratic and that its zeroes must be real for any distinct real numbers $a, b, c$.

If $\alpha$ and $\beta$ are the zeroes of the polynomial $f(x) = x^2 - 5x + k$ such that $\alpha - \beta = 1$, justify the value of $k$.

The graph of a quadratic polynomial $y = ax^2 + bx + c$ is a parabola opening downwards and is completely below the x-axis. Analyze this information to determine the sign of the coefficient 'a' and the nature of the zeroes of this polynomial. Also, comment on the sign of the discriminant ($D = b^2 - 4ac$).

In the quadratic polynomial $p(x) = x^2 - 6x + k$, one zero is twice the other. Analyze this relationship to calculate the value of $k$.

Formulate a cubic polynomial whose zeroes are $2, -3,$ and $4$. Then, create a new cubic polynomial whose zeroes are the reciprocals of the zeroes of the first polynomial. Justify the relationship between the coefficients of the original and the new polynomial.

A student claims that the polynomial $p(x) = x^4 + 5x^2 + 6$ must have four real zeroes because its degree is 4. Critique this statement and justify your reasoning without drawing the graph.

Design a quadratic polynomial $p(x) = ax^2 + bx + c$ such that its graph is a parabola opening downwards and it does not intersect the x-axis. Propose one such polynomial and justify why it meets the given conditions by evaluating its coefficients and discriminant.

If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = 2x^2 - 5x + 7$, apply the relationship between zeroes and coefficients to calculate the values of (i) $\alpha^2 + \beta^2$ and (ii) $\frac{1}{\alpha} + \frac{1}{\beta}$.

Explain with the example $p(x) = x^2 - 2x - 8$ how to find the zeroes of a quadratic polynomial and then verify the relationship between the zeroes and the coefficients.