Binomial Theorem, 3

Category: Algebra

"Published in Newark, California, USA"

Find the middle term for

Solution:

To get the position value of the middle term, which is also the rth term, consider this formula:

where n is the exponent of a binomial. If n is odd number, then the middle term is only one term and if n is even number, then the middle terms are two terms (consecutively). From the given equation above, if n = 11, then the value of r will be equal to

In this case, we need to get the 6th term of the binomial. To get the value of the rth term of (x + y)ⁿ, the formula can be written as

where r > 1. The given formula above must be memorized at all time especially when you are studying about Binomial Theorem. If you don't remember the formula, then you have to use the Pascal's Triangle in order to get the coefficients. Unfortunately, not all people can remember the coefficients of Pascal's Triangle especially if the exponent is high. 

Let's go back to the given problem, if n = 11 and r = 6, then the value of the rth term or 6th term will be equal to

 

 

 

 

Theory - Polynomial Equations, 3

Category: Algebra

"Published in Newark, California, USA"

Form the equation with the following roots:

Solution:

Since there are three given roots, then the degree of a polynomial must be a third degree but in this case, one of the roots has imaginary number, then the degree of a polynomial will not be a third degree.

If the first root is 2, then the factor of a polynomial is (x - 2).

If the next root is -5, then the factor of a polynomial is (x + 5). 

If the last root is 3 + 2i, then we need it's conjugate which is 3 - 2i. Therefore, the factors of a polynomial are (x - 3 - 2i) and (x - 3 + 2i).

Therefore, the equation of a polynomial will be equal to 

 

Theory - Polynomial Equations, 2

Category: Algebra

"Published in Newark, California, USA"

Form the equation with the following roots:

Solution:

Since there are three given roots, then the degree of a polynomial must be a third degree but in this case, two of the roots have irrational numbers or radicals, then the degree of a polynomial will not be a third degree.

If the first root is -5, then the factor of a polynomial is (x + 5).

If the next root is 2 + √3 , then we need it's conjugate which is 2 - √3. Therefore, the factors of a polynomial are (x - 2 - √3) and (x - 2 + √3).

If the last root is 1 + √2, then we need it's conjugate which is 1 - √2. Therefore, the factors of a polynomial are (x - 1 - √2) and (x - 1 + √2). 

Therefore, the equation of a polynomial will be equal to