5.3 Graphs of Polynomial Functions

Solve $\left(4 x - 1\right) \left(x + 6\right) = 0$

$x^{2} - 2 x = 63$

Find the y- and x-intercepts of $g \left(x\right) = \left(x - 2\right)^{2} \left(2 x + 3\right) .$

Find the y- and x-intercepts of the function $f \left(x\right) = x^{4} - 19 x^{2} + 30 x .$

Show that the function $f \left(x\right) = x^{3} - 5 x^{2} + 3 x + 6$ has at least two real zeros between $x = 1$ and $x = 4.$

Show that the function $f \left(x\right) = 7 x^{5} - 9 x^{4} - x^{2}$ has at least one real zero between $x = 1$ and $x = 2.$

Use technology to find the maximum and minimum values on the interval $\left[−1 , 4\right]$ of the function $f \left(x\right) = - 0.2 \left(x - 2\right)^{3} \left(x + 1\right)^{2} \left(x - 4\right) .$

What is the difference between an $x -$ intercept and a zero of a polynomial function $f ?$

Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

If the graph of a polynomial just touches the x-axis and then changes direction, what can we conclude about the factored form of the polynomial?

Find the $x -$ or t-intercepts of the polynomial functions.

$C \left(t\right) = 3 \left(t + 2\right) \left(t - 3\right) \left(t + 5\right)$

Find the $x -$ or t-intercepts of the polynomial functions.

$C \left(t\right) = 2 t \left(t - 3\right) \left(t + 1\right)^{2}$

Find the $x -$ or t-intercepts of the polynomial functions.

$C \left(t\right) = 4 t^{4} + 12 t^{3} - 40 t^{2}$

Find the $x -$ or t-intercepts of the polynomial functions.

$f \left(x\right) = x^{3} + x^{2} - 20 x$

Find the $x -$ or t-intercepts of the polynomial functions.

$f \left(x\right) = x^{3} + x^{2} - 4 x - 4$

Find the $x -$ or t-intercepts of the polynomial functions.

$f \left(x\right) = 2 x^{3} - x^{2} - 8 x + 4$

Find the $x -$ or t-intercepts of the polynomial functions.

$f \left(x\right) = 2 x^{4} + 6 x^{2} - 8$

Find the $x -$ or t-intercepts of the polynomial functions.

$f \left(x\right) = x^{6} - 2 x^{4} - 3 x^{2}$

Find the $x -$ or t-intercepts of the polynomial functions.

$f \left(x\right) = x^{5} - 5 x^{3} + 4 x$

Use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

$f \left(x\right) = x^{3} - 9 x ,$ between $x = 2$ and $x = 4.$

Use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

$f \left(x\right) = - x^{4} + 4 ,$ between $x = 1$ and $x = 3$ .

Use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

$f \left(x\right) = x^{3} - 100 x + 2 ,$ between $x = 0.01$ and $x = 0.1$

Find the zeros and give the multiplicity of each.

$f \left(x\right) = x^{2} \left(2 x + 3\right)^{5} \left(x - 4\right)^{2}$

Find the zeros and give the multiplicity of each.

$f \left(x\right) = x^{2} \left(x^{2} + 4 x + 4\right)$

Find the zeros and give the multiplicity of each.

$f \left(x\right) = \left(3 x + 2\right)^{5} \left(x^{2} - 10 x + 25\right)$

Find the zeros and give the multiplicity of each.

$f \left(x\right) = x^{6} - x^{5} - 2 x^{4}$

Find the zeros and give the multiplicity of each.

$f \left(x\right) = 4 x^{5} - 12 x^{4} + 9 x^{3}$

Find the zeros and give the multiplicity of each.

$f \left(x\right) = 4 x^{4} \left(9 x^{4} - 12 x^{3} + 4 x^{2}\right)$

Use the given information about the polynomial graph to write the equation.

Degree 3. Zeros at $x = –2,$ $x = 1,$ and $x = 3.$ y-intercept at $\left(0 , –4\right) .$

Use the given information about the polynomial graph to write the equation.

Degree 5. Roots of multiplicity 2 at $x = 3$ and $x = 1$ , and a root of multiplicity 1 at $x = –3.$ y-intercept at $\left(0 , 9\right)$

Use the given information about the polynomial graph to write the equation.

Degree 5. Double zero at $x = 1 ,$ and triple zero at $x = 3.$ Passes through the point $\left(2 , 15\right) .$

Use the given information about the polynomial graph to write the equation.

Degree 3. Zeros at $x = −3 ,$ $x = −2$ and $x = 1.$ y-intercept at $\left(0 , 12\right) .$

Use the given information about the polynomial graph to write the equation.

Degree 4. Roots of multiplicity 2 at $x = \frac{1}{2}$ and roots of multiplicity 1 at $x = 6$ and $x = −2.$

y-intercept at $\left(0, 18\right) .$

Use a calculator to approximate local minima and maxima or the global minimum and maximum.

$f \left(x\right) = x^{3} - x - 1$

Use a calculator to approximate local minima and maxima or the global minimum and maximum.

$f \left(x\right) = x^{4} + x$

Use a calculator to approximate local minima and maxima or the global minimum and maximum.

$f \left(x\right) = x^{4} - x^{3} + 1$

Write the polynomial function that models the given situation.

A rectangle has a length of 10 units and a width of 8 units. Squares of $x$ by $x$ units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of $x .$

Write the polynomial function that models the given situation.

A square has sides of 12 units. Squares $x + 1$ by $x + 1$ units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of $x .$

Write the polynomial function that models the given situation.

A right circular cone has a radius of $3 x + 6$ and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is $V = \frac{1}{3} \pi r^{2} h$ for radius $r$ and height $h .$