2.3 Polynomial Equations
Chapter
Chapter 2
Section
2.3
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Solutions 68 Videos

Solve for x.

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

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0.24mins
Q1a

Solve for x.

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

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0.20mins
Q1b

Solve for x.

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

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0.21mins
Q1c

Solve for x.

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

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0.18mins
Q1d

Solve for x.

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

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0.18mins
Q1e

Solve for x.

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

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0.21mins
Q1f

Solve for x.

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

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0.21mins
Q1g

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

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0.43mins
Q2a

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

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0.49mins
Q2b

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

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0.58mins
Q2c

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

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1.13mins
Q2d

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

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1.15mins
Q2e

Determine the real roots of each polynomial equation.

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

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0.26mins
Q3a

Determine the real roots of each polynomial equation.

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

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0.31mins
Q3b

Determine the real roots of each polynomial equation.

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

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0.34mins
Q3c

Determine the real roots of each polynomial equation.

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

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0.29mins
Q3d

Determine the real roots of each polynomial equation.

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

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0.42mins
Q3e

Determine the real roots of each polynomial equation.

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

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0.34mins
Q3f

Determine the real roots of each polynomial equation.

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

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0.40mins
Q3g

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

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

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0.39mins
Q4a

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

y = x^4 - 81x^2

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0.30mins
Q4b

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

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

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0.48mins
Q4c

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

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

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0.43mins
Q4d

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

y = x^4 - 16

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1.04mins
Q4e

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

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

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1.06mins
Q4f

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

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

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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.
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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.
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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.
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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.
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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.
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0.16mins
Q5e

Solve by factoring.

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

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1.36mins
Q6a

Solve by factoring.

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

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0.54mins
Q6b

Solve by factoring.

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

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0.51mins
Q6c

Solve by factoring.

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

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1.34mins
Q6d

Solve by factoring.

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

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1.47mins
Q6e

Solve by factoring.

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

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1.09mins
Q6f

Solve by factoring.

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

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1.36mins
Q6g

Solve by factoring.

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

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2.43mins
Q7a

Solve by factoring.

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

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2.07mins
Q7b

Solve by factoring.

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

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3.36mins
Q7c

Solve by factoring.

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

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4.30mins
Q7d

Solve by factoring.

8x^4 - 64x= 0

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0.46mins
Q7e

Solve by factoring.

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

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2.01mins
Q7f

Solve by factoring.

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

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1.37mins
Q7g

Solve by factoring.

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

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1.11mins
Q8a

Solve by factoring.

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

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3.02mins
Q8b

Solve by factoring.

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

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1.08mins
Q8c

Solve by factoring.

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

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1.40mins
Q8d

Solve by factoring.

x^4 + 13x^3 = -36

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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?

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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.

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3.44mins
Q11

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

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1.13mins
Q12

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

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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?

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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?

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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.
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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?
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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.
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1.30mins
Q16c

Solve. Round answers to one decimal place if necessary.

2(x - 1)^3 = 16

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0.22mins
Q17a

Solve. Round answers to one decimal place if necessary.

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

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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

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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

(b) Determine the other roots of the equation. Justify your answer.

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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.

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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.

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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.

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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.

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2.41mins
Q22

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.

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7.38mins
Q24