MULTIPLYING POLYNOMIALS: Factoring Polynomials
- Factor and graph polynomials
NOTES:
We know how to multiplay polynomials and have seen how graphing and factoring are related.
In this activity we'll combine some of the methods for factoring polynomials.
4 2
3x - 48 = 3(x - 2)(x + 2)(x + 4)
WORKSPACE:
ANS:
25 , x - 5 , 3t(t + 4)(t - 4) , 6(n + 3)(n - 3) , 4p(p + 2)(p - 2) , 5z2(z + 1)(z - 1) , (x + 1)(x - 1) , 4 , (x + 5)(x - 5) , 2 , (x + 4)(x - 4) , 3 , (x + 2)(x - 2) , 4 , 25 , +5 -5 , n + 4 , k - 5 , -4v3 , -2(x + 4)(x - 4) , -4(x + 2)(x - 2) , -3(x + 3)(x - 3) , -x2+1 , -3x2+3 , -4x2+4 , -x2+4 , -2x2+8 , -3x2+12 , 2x(x + 3)(x - 3) , 3x(x + 1)(x - 1) , 3x(x + 2)(x - 2) , x2 + 3 , z - 1 , k3 + 4 , -3x x - 3 , x + 5 , p - 7 , 4k , t , m - 3 , 36 r + 6 , p p , 3 3 , -2t -4 , 2 2 , 3 4 , 3 -5 , -3 -7 , 2p p , 3r2 -5 , 3 5 , -3 , 3 , -1 , -10h2 , 3 3 , +4 -4 , 7 k , +2 +2 , -9 +4 , -2 -3 , -7 -7 , m m , -3h2 +4 , t3 t3
POLYNOMIALS: Problem Set 1
NOTES:
LIGHT:
Light travles at about 2*10^5 miles per second.
Here's our solution.:
Because distance equals rate time time, t = d/r. To divide by the rate, invert the rate and multiply.
1 sec
9*10^7 mi * --------- = 4.5*10^2 mi
2*10^5 mi
Now convert seconds to minutes:
1 min
450 sec * ------ = 7.5 minutes
60 sec
***
RECTANGLE PERIMETER:
The perimeter of a rectangle is 10x-8. The width is x-3. Write a polynomial (in simplest form) for the lenght of the rectangle.
Here's our solution:
l
+---------------+
| |
x-3 | | x-3
| |
+---------------+
l
Since perimeter equals twice the length plus twice the width,
10x - 8 = 2l + 2(x - 3)
10x - 8 = 2l + 2x - 6
8x - 2 = 2l
4x - 1 = length
(That is the letter L not the number one.)
***
POLYNOMIAL FOR FIGURE:
Write a polynomial for the volue of this figure:
+-----+
/ /|
/ / |
/ / +
/ / /
/ / /
/ / / 2x + 3
+-----+ /
2x | | /
+-----+/
4x+2
***
WORKSPACE:
9*100000000
900000000
2*1000000
2000000
***
p = 2l + 2w, so 10x - 8 = 2l + 2(x - 3).
Solve for l (not the number one).
2l + 2(x - 3) = 2l + 2x - 6
2l + 2w = 6
l = -w + 6
*ACK! Now what?!?*
2x(4x + 2)(2x + 3)
2
(8x + 4x)(2x + 3)
3 2 2
16x + 24x + 8x + 12x
3 2
16x + 32x + 12x
***
ANS:
7 1/2 , 4x - 1 , 16x3 + 32x2 + 12x
POLYNOMIALS: Problem Set 2
NOTES:
POLYNOMIAL FOR FIGURE:
Write a polynomial for the volue of a figure consisting of these dimensions:
x, x + 2, 3x - 1
Here's our solution:
TO find the vlume of a rectangular solid, multiply the length times the width times the depth.
x(x + 2)(3x - 1)
2
(x + 2x)(3x - 1)
3 2 2
3x - x + 6x - 2x
3 2
3x + 5x - 2x
***
RECTANGLE PERIMETER:
The perimeter of a rectangle is 14x-6. The width is x + 3. Write a polynomial (in simplest form) for the length of the rectangle.
***
LIGHT:
THe speed of light is 186,000 miles per second. The distance that light can travel in an entire year is known as a light-year. Proxima Centauri, the nearest star to our sun, is 4.25 light-years away. How many miles away is it from our sun? Round your answer to the tenths place and use the powers of twn to write your answer.
Here's our solution:
First let's find the number of seconds in a year.
*got problem right, so I don't need to explain myself ;-) *
***
WORKSPACE:
x(x + 2)(3x - 1)
2
(x + 2x)(3x - 1)
3 2 2
3x - x + 6x - 2x
3 2
3x + 5x - 2x
***
186,000*60*60*24*365
24,929,208,000,000
Rounded to 2.5*10^13.
***
ANS:
3x^3 + 5x^2 - 2x , 6x - 6 , 2.5
POLYNOMIALS: Finger Multiplication
- Excursion
NOTES:
MULTIPLICATION BY NINES:
Pretend that you have caught the dreaded disease: multiplication amnesia!
You have forgotten all of your basic facts about nines...
9 * 9 = ? 9 * 7 = ? 9 * 6 = ? ...
Hold up your hands and number your fingers as shown:
1 2 3 4
\ \ | / 5
\ ||| /
\|||--
7 8 9 10
6 \ | / /
\ ||| /
--|||/
To multiply 9*7, bend down your #7 finger. Now count your fingers to the left of your bent finger. You will come up with the number six.
Then count the fingers right of your bent finger. You will come up with three. The answer is sixty-three.
(9 * 7 = 63)
*this is really cool :-) *
Why did the "nines trick" work? Try it from some other "basic 9's facts".
The 9's finger multiplication works because:
When you multiply nine times any number (from 1 to 9)...
1) The number in the ten's position of the answer is one less than the number, and
2) The sum of the two digits in the answer is always equal to nine.
***
ANOTHER MULTIPLICATION TRICK:
Pretend that you have had a relapse of multiplication amnesia.
You have forgotten all of your basic multiplication facts about numbers larger than five...
7 * 8 = ? 8 * 6 = ? 6 * 7 = ? ...
You multiply 7*8, number your fingers as shown:
6 6
/ \
/ \
---- 7 7 ----
----- 8 8 -----
---- 9 9 ----
--- 10 10 ---
There is a four step procedure for multiplying numbers larger than five on your fingers.
Step 1:
Touch the 7 finger on your left hand to the 8 finger on your right hand. Then, bend the fingers that are below the touching fingers.
Step 2:
Count 10 for each straight finger. In this case, five fingers are straight, so the answer is 50.
Step 3:
Multiply the number of bent fingers on your left hand times the number of bent fingers on your right hand. In this case, the answer is 3 * 2 = 6.
Step 4:
Add the answer from step 2 to the answer from step 3. The final answer is 50 + 6 (or 56).
To make sure that you understand how this trick works use finger multiplication to solve these problems:
9 * 8 = ?
7 * 6 = ?
6 * 9 = ?
Done? Good.
Explanation:
Let's call the two numbers to be multiplied, m and n.
A mathematical description of adding the straight fingers can be written:
7 8
10 * (m - 5) + 10 * (n - 5)
which can be simplified to:
(10 * 2) + (10 * 3)
which is equal to 50.
The next step is to multiply the bent fingers.
A mathematical description of this can be written:
7 8
(10 - m) * (10 - n)
which can be simplified to:
2 * 3
which is equal to 6.
Since the last step adds the results of step 2 and 3, an algebraic description is given by adding the description of steps 2 and 3.
step 2 step 3
10(m - 5) + 10(n - 5) + (10 - m)*(10 - n)
10m - 50 + 10n - 50 + 100 - 10n - 10m + mn
is equal to simply:
m * n
and that's why it works!
***
WORKSPACE:
NONE.
ANS:
NONE.