2.3 Polynomial Equations
Chapter
Chapter 2
Section
2.3
Lectures 10 Videos
Solutions 68 Videos

Solve for x.

 \displaystyle x(x +2)(x - 5) = 0 

0.24mins
Q1a

Solve for x.

 \displaystyle (x - 1)(x - 4)(x + 3) = 0 

0.20mins
Q1b

Solve for x.

 \displaystyle (3x + 2)(x + 9)( x - 2) = 0 

0.21mins
Q1c

Solve for x.

 \displaystyle (x - 7)(3x + 2)(x + 1) = 0 

0.18mins
Q1d

Solve for x.

 \displaystyle (4x - 1)(2x - 3)(x + 8) = 0 

0.18mins
Q1e

Solve for x.

 \displaystyle (2x - 5)(2x + 5)(x - 7) = 0 

0.21mins
Q1f

Solve for x.

 \displaystyle (5x - 8)(x + 3)(2x - 1) = 0 

0.21mins
Q1g

Use the graph to determine the roots of the corresponding polynomial equation. The roots are all integral values.

0.43mins
Q2a

Use the graph to determine the roots of the corresponding polynomial equation. The roots are all integral values.

0.49mins
Q2b

Use the graph to determine the roots of the corresponding polynomial equation. The roots are all integral values.

0.58mins
Q2c

Use the graph to determine the roots of the corresponding polynomial equation. The roots are all integral values.

1.13mins
Q2d

Use the graph to determine the roots of the corresponding polynomial equation. The roots are all integral values.

1.15mins
Q2e

Determine the real roots of each polynomial equation.

(x^2 +1)(x - 4) = 0

0.26mins
Q3a

Determine the real roots of each polynomial equation.

(x^2- 1)(x^2 + 4) = 0

0.31mins
Q3b

Determine the real roots of each polynomial equation.

(3x^2 + 27)(x^2 - 16) = 0

0.34mins
Q3c

Determine the real roots of each polynomial equation.

(x^4 - 1)(x^2 - 25) = 0

0.29mins
Q3d

Determine the real roots of each polynomial equation.

(4x^2 - 9)(x^2 + 16) = 0

0.42mins
Q3e

Determine the real roots of each polynomial equation.

(x^2 + 7x + 12)(x^2 - 49) = 0

0.34mins
Q3f

Determine the real roots of each polynomial equation.

(2x^2 + 5x - 3)(4x^2 - 100) = 0

0.40mins
Q3g

Determine the x-intercepts of the graph of each polynomial function.

y = x^3 - 4x^2 - 45x

0.39mins
Q4a

Determine the x-intercepts of the graph of each polynomial function.

y = x^4 - 81x^2

0.30mins
Q4b

Determine the x-intercepts of the graph of each polynomial function.

y = 6x^3 - 5x^2 - 4x

0.48mins
Q4c

Determine the x-intercepts of the graph of each polynomial function.

y = x^3 + x^2 - 4x - 4

0.43mins
Q4d

Determine the x-intercepts of the graph of each polynomial function.

y = x^4 - 16

1.04mins
Q4e

Determine the x-intercepts of the graph of each polynomial function.

y = x^4- 2x^3 - x^2 + 2x

1.06mins
Q4f

Determine the x-intercepts of the graph of each polynomial function.

y = x^4 - 29x^2 + 100

1.02mins
Q4g

Is this statement true or false? If the statement is false, reword it to make it true.

• If the graph of a quartic function has two x-intercepts, then the corresponding quartic equation has four real roots.
1.45mins
Q5a

Is this statement true or false? If the statement is false, reword it to make it true.

• All the roots of a polynomial equation correspond to the x-int of the graph of the corresponding polynomial function.
0.41mins
Q5b

Is this statement true or false? If the statement is false, reword it to make it true.

• A polynomial equation of degree three must have at least one real root.
0.17mins
Q5c

Is this statement true or false? If the statement is false, reword it to make it true.

• All polynomial equations can be solved algebraically.
0.39mins
Q5d

Is this statement true or false? If the statement is false, reword it to make it true.

• All polynomial equations can be solved graphically.
0.16mins
Q5e

Solve by factoring.

x^3 - 4x^2 - 3x + 18 = 0

1.36mins
Q6a

Solve by factoring.

x^3 - 4x^2 - 7x + 10 = 0

0.54mins
Q6b

Solve by factoring.

x^3 - 5x^2 + 7x - 3 = 0

0.51mins
Q6c

Solve by factoring.

x^3 + x^2 - 8x - 12 = 0

1.34mins
Q6d

Solve by factoring.

x^3 - 3x^2 - 4x + 12 = 0

1.47mins
Q6e

Solve by factoring.

x^3 + 2x^2 - 7x + 4 = 0

1.09mins
Q6f

Solve by factoring.

x^3 - 3x^2 + x + 5 = 0

1.36mins
Q6g

Solve by factoring.

2x^3 + 3x^2 - 5x - 6 = 0

2.43mins
Q7a

Solve by factoring.

2x^3 - 11x^2 + 12x + 9 = 0

2.07mins
Q7b

Solve by factoring.

9x^3 + 18x^2 - 4x - 8 = 0

3.36mins
Q7c

Solve by factoring.

5x^3 - 8x^2 - 27x + 18 = 0

4.30mins
Q7d

Solve by factoring.

8x^4 - 64x= 0

0.46mins
Q7e

Solve by factoring.

4x^4 - 2x^3 - 16x^2 + 8x = 0

2.01mins
Q7f

Solve by factoring.

x^4 - x^3 - 11x^2 + 9x + 18 = 0

1.37mins
Q7g

Solve by factoring.

x^3 - 5x^2 + 8 = -2x

1.11mins
Q8a

Solve by factoring.

x^3 - x^2 = 4x + 6

3.02mins
Q8b

Solve by factoring.

2x^3 - 7x^2 + 10x - 5 = 0

1.08mins
Q8c

Solve by factoring.

x^4 - x^3 = 2x + 4

1.40mins
Q8d

Solve by factoring.

x^4 + 13x^3 = -36

1.48mins
Q8e

The width of a square based storage tank is 3 m less than its height. The tank has a capacity of 20 m^3. If the dimensions are integer values in metres, what are they?

1.25mins
Q10

The passenger section of a train has width 2x - 7, length 2x +3, and height x - 2, with all dimensions in metres. Solve a polynomial equation to determine the dimensions of the section of the train if the volume is 117 m^3.

3.44mins
Q11

Is it possible for a polynomial equation to have exactly one irrational root?

1.13mins
Q12

Is it possible for a polynomial equation to have exactly one non-real root?

1.13mins
Q13

The distance, d, in kilometres, travelled by a plane after t hours can be represented by d(t) = -4t^3 + 40t^2 + 500t, where 0 \leq t \leq 10. How long does the plane take to fly 4088 km?

2.06mins
Q14

A steel beam is supported by two vertical walls. When a 1000-kg weight is placed on the beam, x metres from one end, the vertical deflection, d, in metres, can be calculated using the formula d(x) = 0.0005(x^4-16x^3 + 512x). How far from the end of the beam should the weight be placed for a deflection of 0 m?

4.11mins
Q15

Based on research, the marketing team at Best of U predicts that when the price of a bottle of a new SPF 50 sunscreen is x dollars, the number, D, in hundreds, of bottles sold per month can be modelled by the function D(x) =-x^3 + 8x^2 + 9x + 100

• Graph the function D(x). Write the domain for this situation.
0.31mins
Q16a

Based on research, the marketing team at Best of U predicts that when the price of a bottle of a new SPF 50 sunscreen is x dollars, the number, D, in hundreds, of bottles sold per month can be modelled by the function D(x) =-x^3 + 8x^2 + 9x + 100

• How many bottles are sold per month when the price of each bottle is \$5?
0.44mins
Q16b

Based on research, the marketing team at Best of U predicts that when the price of a bottle of a new SPF 50 sunscreen is x dollars, the number, D, in hundreds, of bottles sold per month can be modelled by the function D(x) =-x^3 + 8x^2 + 9x + 100

• Determine the value(s) of x that will result in sales of 17 200 bottles of sunscreen per month.
1.30mins
Q16c

Solve. Round answers to one decimal place if necessary.

2(x - 1)^3 = 16

0.22mins
Q17a

Solve. Round answers to one decimal place if necessary.

2(x^2 - 4x)^2 - 5(x^2 - 4x) = 3

2.04mins
Q17b

Determine the value of k such that -2 is one root of the equation 2x^3 + (k + 1)x^2 = 4- x^2

0.41mins
Q18a

(a) Determine the value of k such that -2 is one root of the equation 2x^3 + (k + 1)x^2 = 4- x^2

1.29mins
Q18b

Open-top boxes are constructed by cutting equal squares from the corners of cardboard sheet that measure 32 cm by 28 cm. Determine possible dimensions of the boxes if each has a volume of 1920 cm^3.

6.48mins
Q19

A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i = \sqrt{-1}. When the quadratic formula is used and the discriminant is negative, complex numbers result.

(a) Find all the real and complex solutions to x^3 - 27 = 0

(b) Determine a polynomial equation of degree three with roots x = 3 \pm i and x = -4. Is this equation unique? Explain.

2.16mins
Q20

The dimensions of a gift box are consecutive positive integers such that the height is the least integer and the length is the greatest integer. If the height is increased by 1 cm, the width is increased by 2 cm, and the length is increased by 3 cm, then a larger box is constructed such that the volume is increased by 456 cm^3. Determine the dimensions of each box.

4.28mins
Q21

The roots of the equation 6x^3 + 17x^2 - 5x - 6 = 0 are represented by a, b, and c (from least to greatest). Determine an equation whose roots are a + b, \frac{a}{b}, and ab.

Determine the product of all values of k for which the polynomial equation 2x^3 - 9x^2 + 12 x - k = 0 has a double root.