Solving Polynomial Equations in Factored Form

Introduction to polynomial equations in factored form:
In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. (Source: Wikipedia)

Generally polynomial equations can be written as, x^2 - 3x + 9. Polynomial equations in factored form can be written as

(x - 2) (x - 3).

Example Problems for Solving Polynomial Equations in Factored Form
Solving polynomial equations in factored form example problem 1:

Write the given polynomial expression x^2 - 17x + 16 in factored form.

Solution:

Given polynomial expression is x^2 - 17x + 16

First factorize the given expression, we get

(x^2 - 17x + 16) = (x^2 - 16x - x + 16)

Grouping the first two terms and second two terms, we get

= (x^2 - 16x) - (x - 16)

= x (x - 16) - 1 (x - 16)

= (x - 16) (x - 1)


The factors of the given polynomial expression is (x - 16) and (x - 1)

Answer:

The final answer is (x - 16) and (x - 1)

Solving polynomial equations in factored form example problem 2:

Write the given polynomial expression x^2 + 27x + 140 in factored form.

Solution:

Given polynomial expression is x^2 + 27x + 140

First factorize the given expression, we get

(x^2 + 27x + 140) = (x^2 + 20x + 7x + 140)

Grouping the first two terms and second two terms, we get

= (x^2 + 20x) + (7x + 140)

= x (x + 20) + 7 (x + 20)

= (x + 20) (x + 7)


The factors of the given polynomial expression is (x + 20) and (x + 7)

Answer:

The final answer is (x + 20) and (x + 7)

Solving polynomial equations in factored form example problem 3:

Write the given polynomial expression x^2 + 17x - 434 in factored form.

Solution:

Given polynomial expression is x^2 + 17x - 434

First factorize the given expression, we get

(x^2 + 17x - 434) = (x^2 + 31x - 14x - 434)

Grouping the first two terms and second two terms, we get

= (x^2 + 31x) - (14x + 434)

= x (x + 31) - 14 (x + 31)

= (x + 31) (x - 14)


The factors of the given polynomial expression is (x + 31) and (x - 14)

Answer:

The final answer is (x + 31) and (x - 14)

Algebra is widely used in day to day activities watch out for my forthcoming posts on what does algebraic expression mean and algebraic expression examples. I am sure they will be helpful.

Practice Problems for Solving Polynomial Equations in Factored Form
Solving polynomial equations in factored form practice problem 1:

Write the given polynomial expression x^2 + 4x - 32 in factored form.

Answer:

The final answer is (x + 8) and (x - 4)

Solving polynomial equations in factored form practice problem 2:

Write the given polynomial expression x^2 - 7x - 260 in factored form.

Answer:

The final answer is (x - 20) and (x + 13)

Solving polynomial equations in factored form practice problem 3:

Write the given polynomial expression x^2 + 21x + 110 in factored form.

Answer:

The final answer is (x + 10) and (x + 11)

Consistent System of Equations

Introduction to consistent system of equations:

                  Algebra is a subdivision in math, which comprises of vast operations on equations, polynomials, radicals, rational numbers, logarithms, etc. consistent system of equations are contains variable equations which is to be solved. While solving, if the there is at least one solution it is called as consistent system of equations. If there is more than one solution then they are said to be undetermined system of equations.

     
Consistent System of Equations-example Problem:

The goal here is to solve
3x+2y=5
4x+6y=3 for the variables x and y.
Let's start by solving 3x+2y = 5 for the variable x.
Move the 2y to the right hand side by subtracting 2y from both sides, like this:
Now, the equation system reads:
3x = 5-2y
To isolate the x, we have to divide both sides of the equation by the other variables
         around the x on the left side of the equation.
The last step is to divide both sides of the equation by 3 like this:
3x ÷ 3 = x
To divide 5-2y by 3
Divide each term in 5-2y by 3 term by term.
The solution to your equation is:
x =5/3-2/3y
Next, let's solve 4x+6y = 3 for the variable y.
Move the 4x to the right hand side by subtracting 4x from both sides, like this:
From the left hand side:
4x - 4x = 0
The answer is 6y
From the right hand side:
The answer is 3-4x
Now, the equation reads:
6y = 3-4x
To isolate the y, we have to divide both sides of the equation by the other variables
     around the y on the left side of the equation.
The last step is to divide both sides of the equation by 6 like this:
Divide each term in 3-4x by 6 term by term.
The solution to your equation is:
y =1/2-2/3x
Now, plug the earlier result, x=5/3-2/3y, in for x everywhere it occurs in
y=1/2-2/3x.
This gives y=1/2-2/3(5/3-2/3y). Now all we have to do is solve this for y,to have our first solution.
1/2-2/3*(5/3-2/3*y) evaluates to ½ - 10/9 + 4y/9
Y - 4y/9 = -11/18
5y/9 =-11/18
The last step is to divide both sides of the equation by 5/9 like this:
y = - 11/18 * 9/5
The solution to the equation is:
y = -11/10
Lastly, to find the solution for x, we plug this answer for y into the earlier result that
x=5/3-2/3y.
This gives x=5/3-2/3(-11/10).
On, simplifying this.
x= 12/5,
So, the solutions to your equations are:
x= 12/5
y= -11/10
Since this set of equations has one solution, they are consistent system in nature.

Algebra is widely used in day to day activities watch out for my forthcoming posts on solve math word problems and prime number chart up to 10000. I am sure they will be helpful.
 
Consistent System of Equations-practice Problem:

1. Check whether the following set of equation is consistent or inconsistent?
2x+3y=1
3x+2y=2
[Ans: x =4/5, y = -1/5]