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Quadratic in Form Polynomials Answers

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The following system of equations can be used to find the roots of the equation x3 + 72 = 5x2 + 18x.y = x3 + 72y = 5x2 + 18xGraph this system of equations on the graphing calculator.How many intersection points can you see in the default viewing window? 123

A
1
B
2
C
3
2
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Adjust the window so you can find all of the points of intersection for the system of equations.What are the roots of the original polynomial equation? Check all that apply.

A
–6
B
0
C
6
D
–4
E
3
F
8
3

[___], [___], [___]

Answer:

1

4

[___], [___], [___]

Answer:

-4, -0.5, 1

5

Which system of equations can you use to find the roots of the equation 2x3 + 4x2 – x + 5 = –3x2 + 4x + 9?

A
y = 2x3 + x2 + 3x +5y =9
B
y = 2x3 + x2 y = 3x + 14
C
y = 2x3 + 4x2 – x + 5 y = –3x2 + 4x + 9
6

and

Answers:
Part 1:A
Part 2:A
7

A rectangular sheet of metal has identical squares cut from each corner. The sheet is then bent along the dotted lines to form an open box. The volume of the box is 420 in.3.The equation 4x3 – 72x2 + 320x = 420 can be used to find x, the side length of the square cut from each corner.What is the side length of the square that is cut from each corner, to the nearest inch? in.

Answer:

3

8

Explain why the graphing calculator cannot be used to solve or approximate solutions to all polynomial equations.

Answer:

A graphing calculator cannot solve or approximate every polynomial equation because some polynomial equations have complex, nonreal solutions. Graphs in the real Cartesian x-y plane show only real solutions, so complex solutions cannot be found as x-intercepts or intersections; equations with only complex solutions have no x-intercepts.

9

Sun shades are sold in the shape of right isosceles triangles. If the equation represents one shade that shields 64 square feet of area, which system can be used to find the lengths of the legs of the sun shade?

Question illustration
A
and
B
and
C
and
10

The following system of equations can be used to find the roots of the equation x3 + 72 = 5x2 + 18x.y = x3 + 72y = 5x2 + 18xGraph this system of equations on the graphing calculator.How many intersection points can you see in the default viewing window?

A
1
B
2
C
3
11

Which system of equations can you use to find the roots of the equation 2x3 + 4x2 – x + 5 = –3x2 + 4x + 9?

A
y = 2x3 + x2 + 3x +5y =9
B
y = 2x3 + x2 y = 3x + 14
C
y = 2x3 + 4x2 – x + 5 y = –3x2 + 4x + 9
12

[___], [___], [___]

Answer:

-0.5

13

and

Answer not available
14

Did your explanation include the following?

A
Some polynomial equations have complex solutions.
B
Therefore, complex solutions cannot be found by graphing the related system or function in the Cartesian (x-y) plane.
C
Polynomial functions with all complex solutions do not have x-intercepts, and the related system of equations will not intersect.
15

Use the graphing calculator to graph the system. The solutions to the system are [___]. Since the side lengths of a triangle must be positive, the length of one leg of the sun shade is [___] feet.

A
-5.7 or 5.7
B
-8 or 8
C
-11.3 or 11.3
D
-32 or 32
16

Adjust the window so you can find all of the points of intersection for the system of equations.What are the roots of the original polynomial equation? Check all that apply.

A
–6
B
0
C
6
D
–4
E
3
F
8
17

What are the approximate values of the non-integral roots of the polynomial equation?

A
–5.57
B
–1.95
C
0.21
D
1.27
E
4.73
18

5.7811.332

A
5.7
B
8
C
11.3
D
32
19

What are the approximate values of the non-integral roots of the polynomial equation?

A
–5.57
B
–1.95
C
0.21
D
1.27
E
4.73

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