Math 30-1 - Radical, Rational, Polynomial Equations (B1)

Radical Equations

100

1. The graphed functions 

2. The window used. 

3. The calculator function used to find the answer. 

4. The answer.

What are the four requirements for a graphical solution on a test?

200

sqrt(x+4)=7

x=45

300

sqrt(2x+3)=5

x=11

400

sqrt(4x+5)=x

 x=5 

 x=-1 is extraneous.

500

sqrt(2x-7)=x-5

 x=8 

 x=4 is extraneous.

Rational Equations

100

Extraneous values

What is the name for algebraically-derived solutions that are not actual solutions to the equation?

200

5/(x-2)=7

x=1

300

(3x)/(x+4)=2

x=8

400

(4x+3)/(x+2) = 9/5

x=3/11

500

3x-32/(x-4)=2x

 x=8,-4 

Remainder Theorem

100

 P(a) , for some polynomial  P(x) 

What is the remainder of  (P(x))/(x-a) ?

200

The remainder of 

(x^3+2x^2-6x-7)/(x-1)

 P(1) = (1)^3+2(1)^2-6(1)-7 

P(1) = -10

300

The division of 

(x^3+4x^2-3x+5)/(x+2)

(x^2+2x-7) + 19/(x+2)

400

The division of  (x^4+3x^3-5x^2-2x+120)/(x+5) 

(x^3-2x^2-5x+23)+5/(x+5)

500

The remainder of  (5x^5+3x^3+4x^4-1)/(x+3) 

x=-973

Factor Theorem

100

If  P(a) = 0 for some polynomial  P(x) .

When is  (x-a) a factor of  P(x) ?

200

The best choices for  a when factoring  P(x) by factors of  (x-a) .

What are integral factors of the constant term in  P(x) ?

300

The factors of  x^3+5x^2+7x+3.

(x+3)(x+1)^2

400

One factor of 

x^4-x^3-7x^2+x+6

(x-3)

(x+2)

(x-1)

(x+1)

500

One factor of  3x^4+8x^3-40x^2-28x-15 .

 (x+5) 

(x-3)

(3x^2+2x+1)

Polynomial Equations

100

How you can find factors of  P(x)  from the graph y=P(x) .

What are the x-intercepts of the graph  y=P(x) ?

200

Solve 

x^3-19x-30=0

x=5, -2, -3

300

Solve

6x^3-10x^2-4x+8=0

x=1, (1+-sqrt(13))/3

400

Solve 

2x^4+x^3-8x^2-x+6=0

x=3/2,1,-1,-2

500

Solve 

2x^4-7x^3-15x^2+56x-48=0

x=4,-3

Word Problem Grab Bag

100

A kite is secured to a rope that is tied to the ground. A breeze blows the kite so that the rope is taut while the kite is directly above a flagpole that is 30m from where the rope is staked down. Find the altitude of the kite if the rope is 110m long.

The kite is 105.8m from the ground. 

200

A 12m tall tree was broken into two parts. It was found that the height of the cut part was related to the remaining length by the formula x^3+x-10=0.

Determine the length of the cut piece (x).

The length of the cut piece was 2 m.

300

The speed of a tsunami (in  m/s ) is related to the depth of the ocean in meters by the formula

 v=sqrt(9.8d) 

Determine the depth of the ocean when the speed is 117 m/s .

The ocean's depth would be 1396.84m.

400

Lisa can paint a room in 6 hours, whereas her sister Jessica can paint the same room in 4 hours. How long would it take both sisters to paint 3 rooms?

It would take 7 hours and 12 minutes to paint 3 rooms.

500

Kaiden walks 15 km and bikes 120km. If he bikes 20km faster than he walks, and the total trip takes 12 hours, determine Kaiden's walking speed.

Kaiden walks at 2.27 km/h.