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

  

Algebraic Operations - Radicals

Category: Algebra

"Published in Newark, California, USA"

Find the sum of the following:

Solution:

Consider the given equation above

Factor and then rewrite their coefficients into their exponential expression as follows

Take out all the terms that have fourth power and take their fourth root, we have

  
At the third term, there's a denominator which is x² inside the radical sign. We need to rationalize the denominator by multiplying both sides of the fraction by x² in order to eliminate the fraction as follows

Combine similar terms and simplify