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Solutions
56 Videos

State the possible degree of each function, the sign of the leading coefficient, and the number of turning points.

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Q3b

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`

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Q4a

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

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Q4b

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

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Q4c

9, -5, -4

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Q4d

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

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Q5a

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

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Q5b

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

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Q5c

-3, 3, -6, 6

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Q5d

Sketch the graph of
`f(x)=5 (x - 3) (x + 2) (x + 5)`

using the zeros and end behaviours.

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Q6

Determine the equation of the function with zeros at `\pm 1`

and `-2`

, and a y-intercept of `-6`

. Then sketch the function.

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Q7

Describe the transformations that were applied to `y = x^2`

to obtain each of the following functions.

`y = -2(x -1)^2 + 23`

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Q8a

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`

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Q8b

Describe the transformations that were applied to `y = x^2`

to obtain each of the following functions.

`y = x^2 - 8x + 16`

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Q8c

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})
```

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Q8d

Describe the transformations that were applied to `y = x^2`

to obtain each of the following functions.

```
y =40(-7(x - 10))^2 + 9
```

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Q8e

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 25, horizontally compressed by a factor of 5, 6 horizontally translated 3 units to the right

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Q9a

The function `y = x^3`

has undergone each of the following sets of transformations. List three points on the resulting function.

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Q9a

`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

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Q9b

`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

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Q9c

`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

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Q9d

`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

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Q9e

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

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Q9f

Calculate each of the following using long division.

```
\displaystyle
(2x^3 + 5x^2 + 3x - 4) \div (x + 5)
```

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Q10a

Calculate each of the following using long division.

```
\displaystyle
(x^4 + 4x^3 - 3x^2 - 6x -7) \div (x^2 - 8)
```

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Q10b

Calculate each of the following using long division.

```
\displaystyle
(2x^4 -2x^2 + 3x -16) \div (x^3 + 3x^2 + 3x -3)
```

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Q10c

Calculate each of the following using long division.

```
\displaystyle
(x^5 -8x^3 - 7x - 6) \div (x^4 + 4x^3 +4x^2 -x -3)
```

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Q10d

Divide each polynomial by `x + 2`

using synthetic division.

```
\displaystyle
2x^3 + 5x^2 -x -5
```

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Q11a

Divide each polynomial by `x + 2`

using synthetic division.

```
\displaystyle
3x^3 + 13x^2 + 17x + 3
```

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Q11b

Divide each polynomial by `x + 2`

using synthetic division.

```
\displaystyle
2x^4 + 5x^2 - x- 5
```

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Q11c

Divide each polynomial by `x + 2`

using synthetic division.

```
\displaystyle
2x^3 + 4x^2 - 5x -4
```

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Q11d

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`

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Q12a

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`

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Q12b

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`

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Q12c

divisor: `3x^2 + x - 5`

, quotient: `x^4 -4x^3 + 9x -3`

, remainder : `2x -1`

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Q12d

Without dividing, determine the remainder when `x^3 + 2x^2 - 6x + 1`

is divided by `x + 2`

.

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Q13

Factor the polynomial using the factor theorem.

```
\displaystyle
x^3 -5x^2 -22x -16
```

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Q14a

Factor the polynomial using the factor theorem.

```
\displaystyle
2x^3 + x^2 -27x -36
```

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Q14b

Factor the polynomial using the factor theorem.

```
\displaystyle
3x^4 - 19x^3 + 38x^2 -24x
```

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Q14c

Factor the polynomial using the factor theorem.

```
\displaystyle
x^4 + 11x^3 + 36x^2 + 16x -64
```

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Q14d

Factor fully.

```
\displaystyle
8x^3 -10x^2 -17x + 10
```

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Q15a

Factor fully.

```
\displaystyle
2x^3 + 7x^2 -7x -30
```

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Q15b

Factor fully.

```
\displaystyle
x^4 -7x^3 + 9x^2 + 27x -54
```

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Q15c

Factor fully.

```
\displaystyle
4x^4 + 4x^3 - 35x^2 -36x -9
```

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Q15d

Factor the difference of cubes.

```
\displaystyle
64x^3 -27
```

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Q16a

Factor the difference of cubes.

```
\displaystyle
512x^3 - 125
```

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Q16b

Factor the difference of cubes.

```
\displaystyle
343x^3 -1728
```

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Q16c

Factor the difference of cubes.

```
\displaystyle
1331x^3 -1
```

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Q16d

Factor the sum of cubes.

```
\displaystyle
1000x^3 + 343
```

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Q17a

Factor the sum of cubes.

```
\displaystyle
1728x^3 + 125
```

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Q17b

Factor the sum of cubes.

```
\displaystyle
27x^3 + 1331
```

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Q17c

Factor the sum of cubes.

```
\displaystyle
216^3 + 2197
```

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Q17d

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.

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Q18