Trinomial Notes

Trinomial notes is a polynomial with three monomial terms. In trinomial notes, sum of three monomials is said to be  trinomial. For example, 2x5 – 3x2 + 3 is a trinomial . An Algebraic expression of the form axn is called a monomial in x, The sum of two monomials is called a binomial and the sum of three monomials is called a trinomial.  sum of the finite number of a monomials in term x is called a polynomial of x.

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How To Factor trinomial Notes:

Example 1:
Factor   trinomial notes x² + 3x − 10.

Solution.
The binomial factors will have this form:
(x   a)(x   b)
What are the factors of 10?  Let us take that they are 2 and 5:

x² + 3x − 10 = (x   2)(x   5).

We have to know how to choose the signs so that,  coefficient for middle term  will be sum of the outers plus the inners -- will be +3.                                  Choose factors , +5 and −2.

=  x² + 3x − 10

= (x − 2)(x + 5)
When 1 is the coefficient for x², the order of  factors it does not matter.

(x − 2)(x + 5) = (x + 5) (x − 2) are same factors.

Example Problems In trinomial notes:

Example 1:
Find factors for trinomial notes x² − x − 12.

Solution:
We must finding factors for 12 whose algebraic sum will be  coefficient of x is −1.
Choose −4 and + 3:
x² − x − 12 = (x − 4 )(x + 3). are facors

Example 2:

Find factors for trinomial notes    x2 + 5xy + 4y2
Solution:
x2+5xy+4y2

=x2+xy+4xy+4y2

=x(x+y)+4y(x+y)

=(x+y)(x+4y)

Example 3:

Find factors for trinomial notes    2a2 - 7ab + 6b2
Solution:
2a2-7ab+6b2
=2a2-3ab-4ab+6b2

=a(2a-3b)-2b(2a+3b)

=(2a-3b)(a-2b)

Example 4:

Find factors for trinomial notes 4x2 - 13xy + 3y2
Solution:
4x2-13xy+3y2
=4x2-xy-12xy+3y2

=x(4x-y)-3y(4x-y)

=(4x-y)(x-3y)

Acute Angles Pictures

Here we are going to see about the introduction to angles. The angle is referred as a figure which is formed by distribution of  two rays with a common point. This common point is referred as end point. The word angle is from the Latin word angulus which defines the point in corner. There are many types of angles in this we are going to see about the acute angles.

The acute angle is the type of angle which measures the angle between 0 to 90 degree and less than the 90 degree. The normal picture of the acute angle is given as,

Pictures of acute angle:
            The acute angles are referred by using the pictures which measures the angle less then 90 degree. The pictures of the acute angles are given below as examples.
Picture 1:

This is the picture of acute angle which defines that the angle measured in this picture is 48⁰ which is less than 90 degree.

Picture 2:

This is other picture of acute angle which defines that the angle measured in this picture is 28⁰ which is less than 90 degree.

Picture 3:

This is the picture of acute angle which defines that the angle measured in this picture is 37⁰ which is less than 90 degree.

Picture 4:

This is also a picture of acute angle which defines that the angle measured in this picture is 63⁰ which is less than 90 degree.

Picture 5:

This is also a picture of acute angle which defines that the angle measured in this picture is 49⁰ which is less than 90 degree.

Picture 6:

This is also a picture of acute angle which defines that the angle measured in this picture is 72⁰ which is less than 90 degree.

Example problems

Problem 1:
When the angle a-180⁰ is an acute angle. what is the value of a?
Options:
a) 240⁰
b) 300⁰
c) 290⁰
d) 320⁰

Solution:
Acute angle is the angle which measures less than 90 degree or 90 degree.
When 240⁰-180⁰ = 60⁰.
Hence the value of a is 60⁰.

Problem 2:
When the angle y - 90⁰ is an acute angle. what is the maximum value of y?
Options:
a) 170⁰
b) 200⁰
c) 100⁰
d) 540⁰

Solution:
Since the acute angle measures less than 90 degree. The maximum value of y is 170⁰.
Where 200⁰- 90⁰ = 110⁰, since it is greater than 90 degree this is not an acute angle.  
100⁰- 90⁰ = 10⁰, since it is less than 90 degree this is an acute angle but it is the minimum value.

540⁰- 90⁰ = 450⁰, since it is greater than 90 degree this is not an acute angle.  
170⁰- 90⁰ = 80⁰, since it is less than 90 degree this is an acute angle.