Polynomials

The graph of a polynomial $p(x)$ is shown below. Observe the graph to find the zero(s) of the polynomial and verify the result by finding the roots of the expression $p(x) = x^2 - 4$. Based on the graph, what are the zeros of $p(x)$?

How many zeros does the constant polynomial $p(x) = 0$ (the x-axis itself) have?

Based on the graph of $p(x) = -x + 3$, find the value of $x$ for which $p(x) = 0$.

The graph of $y = (x-1)^2$ touches the x-axis at exactly one point. What is the zero of this polynomial?

Find the zeros of $p(x) = x^2 - 4x + 3$ from the graph and verify them by factoring the quadratic.

By observing the graph of $p(x) = x^2 + 4x + 4$, find the zero and verify it using the algebraic expression $(x+2)^2$.

Determine the zeros of the polynomial $p(x) = x^2 - 9$ from the graph and verify them using the identity $a^2 - b^2$.

The velocity of a particle is given by $v(t) = t^2 - t - 12$. The particle comes to a momentary rest when $v(t) = 0$. Using the graph of the velocity function, find the time $t > 0$ when the particle is at rest.

A valley's cross-section is modeled by $v(x) = x^2 + 2x - 8$. The x-axis represents the sea level. Find the width of the valley at sea level by identifying the distance between the zeros from the graph and verifying algebraically.

A signal wave is represented by $f(x) = x^2 + 4$. A technician is looking for the 'grounding points' (zeros). Looking at the graph, what can be concluded about the zeros of this polynomial in the real number system?

The product of the zeroes of the polynomial $2x^2 - 8$ is:

If $\alpha$ and $\beta$ are the zeroes of $x^2 - 4x + 1$, then the value of $\alpha + \beta - \alpha\beta$ is:

If the product of the zeroes of the polynomial $x^2 - 3x + k$ is $-10$, then $k$ is:

In the polynomial $ax^2 + bx + c$, the sum of the zeroes is $0$ if:

A quadratic polynomial whose zeroes are $3 + \sqrt{2}$ and $3 - \sqrt{2}$ is:

If one zero of the quadratic polynomial $x^2 - 4x + 1$ is $2 + \sqrt{3}$, then the product of its zeroes is:

If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 - mx + n$, then the value of $\alpha^2 + \beta^2$ is:

If $\alpha, \beta$ are zeros of $ax^2+bx+c$, find the value of $\frac{1}{\alpha^3} + \frac{1}{\beta^3}$.

If $\alpha, \beta$ are the zeros of the polynomial $x^2 - 2x - 8$, find the value of $\alpha^3 + \beta^3$.

If $\alpha, \beta$ are the zeros of the polynomial $x^2 + px + q$, find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} + 2$.

If $x - 1$ is a factor of $x^2 + x + k$, then $k$ is:

What is the degree of $x^2 + x + x^3 + 1$?

The product of the zeroes of $4x^2 - 1$ is:

If the sum of zeroes of $x^2 - kx + 6$ is 5, then $k$ is:

If the product of two zeros of the cubic polynomial $f(x) = x^3 - 5x^2 - 2x + 24$ is $12$, then find the third zero of the polynomial.

If the zeros of the quadratic polynomial $ax^2 + bx + c$ ($a \neq 0$) are both negative, what can be concluded about the signs of the coefficients $a, b,$ and $c$?

If $\alpha$ and $\beta$ are zeros of $x^2 - x - 2$, find the value of $\alpha^2\beta + \alpha\beta^2$.

If $\alpha, \beta$ are zeros of $x^2 - x - 2$, find the polynomial whose zeros are $2\alpha + 1$ and $2\beta + 1$.

If $\alpha$ and $\beta$ are the zeros of $x^2 + kx + 12$ such that $\alpha - \beta = 1$, then $k^2$ equals:

Find the sum of the reciprocals of the zeros of the polynomial $f(x) = x^4 + 3x^3 - 2x^2 + 5x - 6$.

If a polynomial $p(x)$ has zeroes at $x=1, x=2, x=3$, how many times does its graph intersect the $x$-axis?

In the graph of a polynomial $y = p(x)$, the $y$-intercept is found by calculating $p(k)$ for what value of $k$?

If the graph of a polynomial $p(x)$ is a line segment that starts at $(1, 1)$ and ends at $(5, 5)$, how many zeroes does it have?

The maximum number of zeroes of a polynomial of degree 5 is:

A quadratic polynomial $f(x) = ax^2 + bx + c$ has its vertex located in the first quadrant of the Cartesian plane. If the graph of $y = f(x)$ passes through the origin $(0, 0)$, how many distinct real zeroes does the polynomial have?

If the graph of $y = ax^2 + bx + c$ has no $x$-intercepts and $a > 0$, then the entire graph lies:

The number of points of intersection of the graph $y = p(x)$ with the $x$-axis gives the number of ________ of the polynomial $p(x)$.

If the graph of $y = ax^2 + bx + c$ intersects the x-axis at $(-3, 0)$ and $(5, 0)$, what is the x-coordinate of the vertex?

A polynomial $p(x)$ has degree $n$. The graph of $y = p(x)$ can intersect any horizontal line $y = k$ at most how many times?

If $p(x)$ is a quadratic polynomial with real zeroes, and we increase the constant term $c$ significantly, what happens to the number of x-intercepts if $a > 0$?

If the product of zeroes of $x^2 - 3x + k$ is -10, then $k$ is:

A quadratic polynomial whose zeroes are 5 and -5 is:

If $\alpha$ and $\beta$ are the zeroes of $x^2 - 1$, then $\alpha + \beta$ is:

Find the product of zeroes of $x^2 + 10x + 25$.

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = x^2 - 7x + k$ such that $\alpha^2 - \beta^2 = 7$, then the value of $k$ is:

If the sum of the zeroes of the quadratic polynomial $p(x) = (k-2)x^2 + (k+3)x + (3k-1)$ is $-2$, then the product of its zeroes is:

If $\alpha$ and $\beta$ are zeroes of $f(x) = x^2 + x - 2$, find the value of $\frac{1}{\alpha} - \frac{1}{\beta}$.

If $\alpha, \beta$ are zeroes of $x^2 - 8x + k$ and $\alpha^2 + \beta^2 = 40$, find $k$.

If $\alpha, \beta, \gamma$ are zeroes of $2x^3 - x^2 - 5x - 2$, find the value of $\alpha\beta\gamma$.

If the sum of zeroes of $kx^2 + 2x + 3k$ is equal to their product, find $k$.