Math Examples of Polynomials

Given expression is (7x – 12)³, and to solve this we can use the formula given as:  
(a – b)³ = a³ -3a²b + 3ab² –b³ = a³ -3ab(a –b)–b³;
Putting the value in the given formula. i.e.
a = 7x;
b = 12;
(a – b)³ = a³ -3ab(a –b)–b³
= (7X)³  - 3 * 7x * 12( 7x – 12) – (12)³
= 343X³  - 252x( 7x – 12) – (12)³
On further solving the equation we get
= 343X³  - 1764x² + 3024x – 1728
This is the required solution of the given equation
(7x – 12)³  =   343X³  - 1764x² + 3024x – 1728. 

We know given equation is ( z + 8xy⁵)² and we can solve this equation using the formula given as:
(a + b)² = a² +2ab +b²
Putting the value in the given formula. i.e.
a = z;
b = 8xy⁵;
(a + b)² = a² +2ab +b²,
= (z²) + 2 * (z)(8xy⁵) + (8xy⁵)²,
= z²  + 16xy⁵z + 64x²y¹⁰,
This is the required solution of the given equation:
( z + 8xy⁵)²  =  z²  + 16xy⁵z + 64x²y¹⁰.

We have (3x – 5)³, as we know that this equation is solved by the formula i.e given as: 
(a – b)³ = a³ -3a²b + 3ab² –b³ = a³ -3ab(a –b)–b³;
Putting the value in the given formula. i.e.
a = 3x;
b = 5;
(a – b)³ = a³ -3ab(a –b)–b³
= (3X)³  - 3 * 3x * 5( 3x – 5) – (5)³
= 27X³  - 45x( 3x – 5) – (5)³
On further solving the equation we get
= 27X³  - 135x² + 225x – 625
This is the required solution of the given equation:
(3x – 5)³  =  27X³  - 135x² + 225x – 625.

Given expression is (x – 8)³, to solve this equation we have to use the given formula. i.e.  
(a – b)³ = a³ -3a²b + 3ab² –b³ = a³ -3ab(a –b)–b³;
Putting the value in the given formula. i.e.
a = x;
b = 8;
(a – b)³ = a³ -3ab(a –b)–b³
= (X)³  - 3 * x * 8( x – 8) – (8)³
= X³  - 24x( x – 8) – (8)³
On further solving the equation we get
= x³  - 24x² + 192x – 512
This is the required solution of the given equation.
(x – 8)³  =  X³  - 24x² + 192x – 512.

We know that given equation i.e. (yz² – 4) ( y² z⁴ + 4yz² + 16) is solved by the given formula. i.e.
(a - b)(a² + ab + b²) = a³ - b³;
Putting the value in the given formula. i.e.
a = yz;
b = 16;
(a - b)(a² + ab + b²) = a³ - b³
= (yz²) - (3) * (yz²)²  * (yz²)(4) – (4)²
= (yz²)³ – (4)³
On further solving the equation we get
= (y³z⁶) - 64 
This is the required solution of the given equation:
(yz² – 4) ( y² z⁴ + 4yz² + 16) =  (y³z⁶) – 64.

Before going through multiplication method we just follow some steps. i.e.
Step 1: Multiply two Polynomials:
           Multiply each term in one polynomial by each term in the other polynomial.
Step2:- Then add them together, and simplify the equation if needed.
Now we have to follow these steps:
= (5) (2x + 9).
Multiply 5 by all terms of polynomials.
 5 (2x + 9),
= 10x + 45,
10x + 45; this is required solution.

Before going through multiplication method we just follow some steps. i.e.
Step1: Multiply the monomial by polynomial:
Multiply each term in one polynomial by each term in the other polynomial.
Step2:- then add them together, and simplify the equation if needed.
Now we have to follow these steps:
 = -10x³ (xyz – 4a + 7xy),
Multiply -9x³ by all terms of polynomial.
On multiplying we get
 = (-10x⁴yz + 40x³a - 70x⁴y),
Now we have to combine all like term if present in the equation. In this equation there is no like term.
 = (-10x⁴yz + 40x³a - 70x⁴y)
We get (-10x⁴yz + 40x³a - 70x⁴y) multiplicative polynomial.

Before going through multiplication method we just follow some steps. i.e.
Step1: To multiply monomial by Polynomials:
Multiply each term in one polynomial by each term in the other polynomial.
Step2:- then add them together, and simplify the equation if needed.
Now we have to follow these steps:
= (6) (2xz – 5a + 3xy),
Multiply 6 by all terms of polynomials.
 On multiplying we get:
 = (12xz – 30a + 18xy),
Now we have to combine all like terms if present in the equation.
After solving we get ((12xz – 30a + 18xy) multiplicative monomial by a polynomial.

Use the steps given below to solve the given question: 
Step 1: Multiply monomial by polynomial:
Multiply each term in one polynomial by each term in the other polynomial.
Step2:- Then add them together, and simplify the equation if needed.
Now we have to follow these steps:
 = (4y) (6x – 9y + 15)
Multiply ‘4y’ by all terms of polynomials.
On multiplying we get
= (24xy – 36y² + 60y),
Now we have to combine all like terms, if present in the equation.
= (24xy – 36y² + 60y),
After solving we get (24xy – 36y² + 60y) multiplicative monomial by a polynomial.

Follow the given steps in order to solve the problem: 
Step 1: Multiply two Polynomials:
           Multiply each term in one polynomial by each term in the other polynomial.
Step 2:- Then add them together, and simplify the equation if needed.
 Now we have to follow these steps:
 = (4xy) (5x – 4y + 7),
Multiply ‘4xy’ by all terms of polynomial.
On multiplying we get:
 = (20x²y -16xy² +28xy),
Now we have to combine all like terms if present in the equation.
In the given equation there is no like term so:
 = (20x²y -16xy² +28xy),
After solving we get this polynomial.

We can reformat the way to write Numbers by means of the exponents. It is at times easier to use the scientific notation of a number than the number itself when it is very large or small.
The way to write a number in scientific notation is clear - cut: write [number’s first digit] [decimal Point] [all the remainder of the number’s digits] * [10 appropriately raised to the power of number of digits].
Thus, we can write 8363 in scientific notation as:       
8.363 x 10³.

When we change a number from scientific notation to its decimal representation, we first look at the power of the number 10. In this case, it is a negative number. As the exponent of the given number is - 10, we will have to move the decimal Point 10 places to the left to express it in Decimal Notation. It is equivalent to multiplying 2.14 with 1 / 10, 000, 000, 000.
Thus the decimal representation of 2.14 x 10 ^-10 is 0.000000000214. Note that this is quite a small number.
Thus, whenever a number is too large or too small, we require to represent is in smatter terms with the help of the Scientific Notations.

When we convert a number from scientific notation to the Decimal Notation, we first look at the exponent of the number 10. In this case, it is a positive quantity. As the exponent of the given number is 12, we will have to move the decimal Point 12 places to the right to express it in Decimal Notation. It is equivalent to multiplying 1.362 with 1, 000, 000, 000, 000.
Thus the decimal representation of 1.362 x 10 ¹² is 1, 362, 000, 000, 000, this is quite a large number.

It is occasionally easier to use the scientific notation of a number than the number itself when it is very large or small.
The way to write a number whose magnitude is less than one in scientific notation is: write [number’s first digit] [decimal Point] [all the remainder of the number’s digits] * [10 appropriately raised to the power of number of digits].
Thus, we can write 0.7261 in scientific notation as:           
7. 261 x 10 ^-1.