State the possible degree of each function, the sign of the leading coefficient, and the number of turning points.
For the following, write the equations of three cubic functions that have the given zeros and belong to the same family of functions.
-3, 6, 4
For the following, write the equations of three cubic functions that have the given zeros and belong to the same family of functions.
5, -1, -2
For the following, write the equations of three cubic functions that have the given zeros and belong to the same family of functions.
0, -1, 9, 10
For the following, write the equations of three cubic functions that have the given zeros and belong to the same family of functions.
9, -5, -4
For the following, write the equations of three quartic functions that have the given zeros and belong to the same family of functions.
-6, 2, 5, 8
For the following, write the equations of three quartic functions that have the given zeros and belong to the same family of functions.
4,-8, 1, 2
For the following, write the equations of three quartic functions that have the given zeros and belong to the same family of functions.
-7, 2, 3
For the following, write the equations of three quartic functions that have the given zeros and belong to the same family of functions.
-3, 3, -6, 6
Sketch the graph of
f(x)=5 (x - 3) (x + 2) (x + 5)
using the zeros and end behaviours.
Determine the equation of the function with zeros at \pm 1
and -2
, and a y-intercept of -6
. Then sketch the function.
Describe the transformations that were applied to y = x^2
to obtain each of the following functions.
y = -2(x -1)^2 + 23
Describe the transformations that were applied to y = x^2
to obtain each of the following functions.
y = (\frac{12}{13}(x + 9))^2 - 14
Describe the transformations that were applied to y = x^2
to obtain each of the following functions.
y = x^2 - 8x + 16
Describe the transformations that were applied to y = x^2
to obtain each of the following functions.
y =(x + \frac{3}{8})(x + \frac{3}{7})
Describe the transformations that were applied to y = x^2
to obtain each of the following functions.
y =40(-7(x - 10))^2 + 9
The function y = x^3
has undergone each of the following sets of transformations. List three points on the resulting function.
The function y = x^3
has undergone each of the following sets of transformations. List three points on the resulting function.
The function y = x^3
has undergone each of the following sets of transformations. List three points on the resulting function.
reflected in the y-axis, horizontally stretched by a factor of 7, vertically translated 19 units down
The function y = x^3
has undergone each of the following sets of transformations. List three points on the resulting function.
reflected in the x-axis, vertically compressed
by a factor of \frac{6}{11}
, horizontally translated 11
5 units to the left, vertically translated
16 units up
The function y = x^3
has undergone each of the following sets of transformations. List three points on the resulting function.
vertically stretched by a factor of 100, horizontally stretched by a factor of 2, vertically translated 14 units up
The function y = x^3
has undergone each of the following sets of transformations. List three points on the resulting function.
eflected in the y-axis, vertically translated 45 units down
The function y = x^3
has undergone each of the following sets of transformations. List three points on the resulting function
reflected in the x-axis, horizontally
compressed by a factor of \frac{7}{10}
, horizontally
translated 12 units to the right, vertically translated 6 units up
Calculate each of the following using long division.
\displaystyle
(2x^3 + 5x^2 + 3x - 4) \div (x + 5)
Calculate each of the following using long division.
\displaystyle
(x^4 + 4x^3 - 3x^2 - 6x -7) \div (x^2 - 8)
Calculate each of the following using long division.
\displaystyle
(2x^4 -2x^2 + 3x -16) \div (x^3 + 3x^2 + 3x -3)
Calculate each of the following using long division.
\displaystyle
(x^5 -8x^3 - 7x - 6) \div (x^4 + 4x^3 +4x^2 -x -3)
Divide each polynomial by x + 2
using synthetic division.
\displaystyle
2x^3 + 5x^2 -x -5
Divide each polynomial by x + 2
using synthetic division.
\displaystyle
3x^3 + 13x^2 + 17x + 3
Divide each polynomial by x + 2
using synthetic division.
\displaystyle
2x^4 + 5x^2 - x- 5
Divide each polynomial by x + 2
using synthetic division.
\displaystyle
2x^3 + 4x^2 - 5x -4
The divisor was divided into another polynomial, resulting in the given quotient and remainder. Determine the dividend.
divisor: x -9
, quotient: 2x^2 + 11x -8
, remainder : 3
The divisor was divided into another polynomial, resulting in the given quotient and remainder. Determine the dividend.
divisor: 4x +3
, quotient: x^3 -2x + 7
, remainder : -4
The divisor was divided into another polynomial, resulting in the given quotient and remainder. Determine the dividend.
divisor: 3x -4
, quotient: x^3 + 6x^2 -6x -7
, remainder : 3x^2 + x - 5
The divisor was divided into another polynomial, resulting in the given quotient and remainder. Determine the dividend.
divisor: 3x^2 + x - 5
, quotient: x^4 -4x^3 + 9x -3
, remainder : 2x -1
Without dividing, determine the remainder when x^3 + 2x^2 - 6x + 1
is divided by x + 2
.
Factor the polynomial using the factor theorem.
\displaystyle
x^3 -5x^2 -22x -16
Factor the polynomial using the factor theorem.
\displaystyle
2x^3 + x^2 -27x -36
Factor the polynomial using the factor theorem.
\displaystyle
3x^4 - 19x^3 + 38x^2 -24x
Factor the polynomial using the factor theorem.
\displaystyle
x^4 + 11x^3 + 36x^2 + 16x -64
Factor fully.
\displaystyle
8x^3 -10x^2 -17x + 10
Factor fully.
\displaystyle
2x^3 + 7x^2 -7x -30
Factor fully.
\displaystyle
x^4 -7x^3 + 9x^2 + 27x -54
Factor fully.
\displaystyle
4x^4 + 4x^3 - 35x^2 -36x -9
Factor the difference of cubes.
\displaystyle
64x^3 -27
Factor the difference of cubes.
\displaystyle
512x^3 - 125
Factor the difference of cubes.
\displaystyle
343x^3 -1728
Factor the difference of cubes.
\displaystyle
1331x^3 -1
Factor the sum of cubes.
\displaystyle
1000x^3 + 343
Factor the sum of cubes.
\displaystyle
1728x^3 + 125
Factor the sum of cubes.
\displaystyle
27x^3 + 1331
Factor the sum of cubes.
\displaystyle
216^3 + 2197
a) Factor the expression x^6 - y^6
completely by
treating it as a difference of squares.
b) Factor the same expression by treating it as a difference of cubes.
c) Explain any similarities or differences in your final results.