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Lectures
9 Videos

1 Introduction to Rational Zero Theorem

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3.44mins

1 Introduction to Rational Zero Theorem

2 Introduction to Intermediate Value Theorem IVT

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5.50mins

2 Introduction to Intermediate Value Theorem IVT

3 Factoring Deg 4 Polynomial Example

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2.43mins

3 Factoring Deg 4 Polynomial Example

4 Factoring Deg 4 Polynomial with Irrational Roots

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4.48mins

4 Factoring Deg 4 Polynomial with Irrational Roots

5 Complete Factorization Theorem

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2.46mins

5 Complete Factorization Theorem

6 Finding the 7olynomial from given root information ex2

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2.29mins

6 Finding the 7olynomial from given root information ex2

6 Finding the polynomial from given root information ex1

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4.06mins

6 Finding the polynomial from given root information ex1

8 Pattern of Coefficients and Roots for Higher Powers

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3.59mins

8 Pattern of Coefficients and Roots for Higher Powers

9 Rational Zero Theorem Proof

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6.56mins

9 Rational Zero Theorem Proof

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

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0.49mins

Q2b

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0.58mins

Q2c

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1.13mins

Q2d

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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