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Solutions
36 Videos

Solve each of the following equations by factoring.

`x^4-16x^2 + 75 = 2x^2 -6`

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2.43mins

Q1a

Solve each of the following equations by factoring.

`2x^2 + 4x - 1 = x+ 1`

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0.55mins

Q1b

Solve each of the following equations by factoring.

```
\displaystyle
4x^3 -x^2 -2x + 2 = 3x^3 -2(x^2 - 1)
```

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1.06mins

Q1c

Solve each of the following equations by factoring.

```
\displaystyle
-2x^2 + x -6 = -x^3 + 2x -8
```

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0.34mins

Q1d

Solve the equation algebraically, and check your solution graphically:

```
\displaystyle
18x^4 -53x^3 + 52x^2 - 14x - 8 = 3x^4 -x^3+2x - 8
```

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Q2

a) Write the equation of a polynomial `f(x)`

that has a degree of `4`

, zeros at `x = 1, 2, -2`

, and `-1`

, and has a `y`

-intercept of `4`

.

b) Determine the values of `x`

where `f(x) = 48`

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Q3

An open-topped box is made from a rectangular piece of cardboard, with dimensions of `24`

cm by `30`

cm, by cutting congruent squares from each corner and folding up the sides. Determine the dimensions of the squares to be cut to create a box with a volume of `1040 cm^3`

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2.37mins

Q4

Between 1985 through 1995, the number of home computers, in thousands, sold in Canada is estimated by `C(t) = 0.92(t^3 + 8t^2+40t + 400)`

,

where t is in years and `t = 0`

for `1985`

.

a) Explain why you can use this model to predict the number of home computes sold in 1993, but not to predict sales in 2005.

b) Explain how to find when the number of home computers sales in Canada reached 1.5 million, using this model.

c) In what year did home computer sales reach 1.5 million?

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Q5

For the number line given, write an inequality with both constant and linear terms of each side that has the corresponding solution.

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Q6a

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Q6b

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0.17mins

Q6c

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Q6d

Solve the inequalities algebraically. State your answers using interval notation.

```
\displaystyle
2(4x - 7) > 4(x + 9)
```

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Q7a

Solve the inequalities algebraically. State your answers using interval notation.

```
\displaystyle
\frac{x -4}{5} \geq \frac{2x + 3}{2}
```

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Q7b

Solve the inequalities algebraically. State your answers using interval notation.

```
\displaystyle
-x + 2 > x -2
```

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0.26mins

Q7c

Solve the inequalities algebraically. State your answers using interval notation.

```
\displaystyle
5x -7 \leq 2x + 2
```

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Q7d

Solve the inequalities algebraically. State your answers using set notation.

```
\displaystyle
-3 < 2x + 1 < 9
```

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0.51mins

Q8a

Solve the inequalities algebraically. State your answers using set notation.

```
\displaystyle
8 \leq -x + 8 \leq 9
```

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Q8b

Solve the inequality.

```
\displaystyle
6 + 2x \geq 0 \geq -10 + 2x
```

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Q8c

Solve the inequalities algebraically. State your answers using set notation.

```
\displaystyle
x + 1 < 2x + 7 < x + 5
```

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Q8d

A phone company offers two options. The first plan is an unlimited calling plan for $34.95 a month. The second plan is a $20.95 monthly free plus $0.04 a minute for call time.

a) When is the unlimited plan a better deal?

b) Graph the situation to confirm your answer from part a).

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Q9

Solve for x.

```
\displaystyle
(x + 1)(x-2)(x + 3)^2 < 0
```

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1.03mins

Q10a

Solve for x.

```
\displaystyle
\frac{(x-4)(2x + 3)}{5} \geq \frac{2x + 3}{5}
```

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Q10b

Solve for x.

```
\displaystyle
-2(x - 1)(2x + 5)(x - 7) > 0
```

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Q10c

Solve for x.

```
\displaystyle
x^3 + x^2 -21x + 21 \leq 3x^2 - 2x + 1
```

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0.48mins

Q10d

Determine algebraically where the intervals of the function are positive and negative.

```
\displaystyle
f(x) = 2x64 -2x^3 -32x^2 -40x
```

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Q11

Solve the following inequality using graphing tech.

```
\displaystyle
x^3 -2x^2 + x- 3 \geq 2x^3 + x^2 -x + 1
```

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1.02mins

Q12

Hundreds of thousands of cubic metres of wood are harvested each year. The function

**requires a graphing device**

```
\displaystyle
f(x) = 1135x^4 -8197x^3 + 15 868x^2 - 2157x + 176 608, 0 \leq x \leq 4,
```

models the volume harvested, in cubic metres, from 1993 to 1997. Estimate the intervals (in years and months) when less than `185 000 m^3`

were harvested.

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3.12mins

Q13

For each of the following functions, determine the average rate of change in `f(x)`

from `x =2`

to `x = 7`

, and estimate the instantaneous rate of change at `x = 5`

.

`f(x) = x^2 -2x + 3`

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Q14a

For each of the following functions, determine the average rate of change in `f(x)`

from `x =2`

to `x = 7`

, and estimate the instantaneous rate of change at `x = 5`

.

`h(x) =(x - 3)(2x + 1)`

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1.12mins

Q14b

For the following functions, determine the average rate of change in `f(x)`

from `x =2`

to `x = 7`

,

and estimate the instantaneous rate of change at `x = 5`

.

`g(x) =2x^3 - 5x`

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Q14c

For the following functions, determine the average rate of change in `g(x)`

from `x =2`

to `x = 7`

,

and estimate the instantaneous rate of change at `x = 5`

.

`g(x) = -x^4 +2x^2 -5x + 1`

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0.44mins

Q14d

Given the following graph, determine the intervals of `x`

where the instantaneous rate of change is positive, negative and zero.

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Q15

The height in metres of projectile is modelled by function `h(t) = -5t^2 + 24`

, where `t`

is the time seconds.

a) Find the point when the object hits the ground.

b) Find the average rate of change from the point when the projectile is launched (`t=0`

) to the point which it hits the ground.

c) Estimate the object's speed at the point of impact.

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Q16

Consider the function `f(x) = 2x^3 + 3x -1`

.

a) Find the average rate of change from `x = 3`

to `x = 3.0001`

.

b) Find the average rate of change from `x = 2.9999`

to `x =3`

c) Estimate the instantaneous rate of change at `x =3`

using your answer above.

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2.08mins

Q17

The incidence of lung cancer in Canadians per 100 000 people is shown below.

a) Use regression to determine a cubic function to represent the curve of best fit for both the male and female data.

b) According to your models, when will more females have lung cancer than males?

c) Was the incidence of lung cancer changing at a faster rte in the male or female population during the period from 1975 to 2000? Justify your answer.

d) Was the incidence of lung cancer changing at a faster rate in the male of female population in 1998? Justify your answer.

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Q18