## Wednesday, May 30, 2012

When given the lenghts of all 3 sides of a triangle, how do we determine if the triangle is right, acute or obtuse?

Recall that for a right triangle with sides of length a,b and c (with c being the hypotenuse),
a^2 + b^2 = c^2

So a triangle with sides 7, 24 and 25 is a right triangle

7^2 + 24^2 = 25^2
49 + 576 = 625

If a^2 + b^2 < c^2, the triangle is obtuse.

A triangle with sides 4, 6 and 10 is an example of an obtuse triangle

4^2 + 6^2 < 10^2

16 + 36 < 100

52 < 100

If a^2 + b^2 > c^2, the triangle is acute.

A triangle with sides 7, 8, 9 is an acute triangle

7^2 + 8^2 > 9^2

49 + 64 > 81

113 > 81

1. Nice.The problem is explained in simpler step by step procedure.If possible please explain me "If the zeroes of the polynomial x(cube of x) -- 3x)Squared) + x + 1 are a -- b, a, a + b, find a and b http://youtu.be/EF2t23amMiE

2. Hi Sree, thanks for the comment! In the example you give, we can use the rational root theorem to determine the possible rational roots of x^3 - 3x^2 + x + 1.

Take all the factors of the coefficient of the last term of 1 and divide by all the factors of the coefficient of the x^3 term of 1. Obviously the only factors of each term are 1. Now we take +/- this quotient, so all the possible rational roots are +/- 1.

Now we can use synthetic division.

Write down all the coefficients of the terms of the polynomial and 1 in the upper left corner, since we are seeing if 1 is a zero

1| 1 -3 1 1

bring down the first 1 and multiply by the 1 in the upper left. That result is written under the -3 and added. The result of the addition is -2 and is written directly underneath

1| 1 -3 1 1
1
1 -2

Now we take -2 and multiply by 1 to get -2 and write that under the 1. Add theses together to get -1 and write directly underneath

1| 1 -3 1 1
1 -2
1 -2 -1

Now take the -1 and multiply by 1, which is -1 and write under the 1. Add these and you get 0.

1| 1 -3 1 1
1 -2 -1
1 -2 -1 0

Since we get 0, we know 1 is a zero of the polynomial. The other numbers we have left (1, -2, -1) are the coefficients of the resulting polynomial from the division. The exponents of each term are reduced by 1 from the division. So the resultant polynomial is x^2 -2x - 1.

Set this equal to 0 and use the quadratic formula to get the other zeros of the polynomial. Recall that if a polynomial is of the form ax^2 + bx + c = 0, then

x= [-b +/- sqrt(b^2- 4ac)]/2a

In our problem a = 1, b = -2, c = -1

Therefore,

x = [2 +/- sqrt(4 - 4(1)(-1)]/2

x = [2 +/- sqrt(8)]/2

x = [2 +/- 2sqrt(2)]/2

x = 1 +/- sqrt(2)

So if the answer is in a, a + b, a - b

a = 1, b = sqrt(2)

3. Sorry that the numbers did not all line up in my synthetic division