Polynomials Definition and Concepts

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Polynomials Definition

A polynomial is an algebraic expression in which the variables involving are having non-negative integral powers. 

Examples- 2x³+5x+2, 8y³+5xy+x-2 etc.

Ok, lets now define the term Algebraic Expressions, variables and constants. It is very important for understanding the Polynomials definition.

Variables

A symbol (x,y,z,a,b,g,h etc.) that can be assigned different numerical values.

Example- Area of a triangle = ½×base(b)×height(h), Here ‘b’ and ‘h’ can have different numerical values. So, ‘b’ and ‘h’ are variables. 

Constants

 Symbols having fixed numerical values are called constants. 

For Example- 2, 3, 1/2, 22/7, etc.

Algebraic Expressions

When we combine the variables and constants along with operations +, -, ×, and ÷  (few or all), the result obtained is called an algebraic expression.

Example- 2x³+5x+2, 

Note- This above expression contains 3 terms (Home Work- Name these)

Coefficients

The numerical values associated with the variables in a polynomial are called coefficients. For example- In the polynomial,  2x³+5x²+x+2 the coefficients of x³, x², x are 2, 5 and 1 respectively whereas 2 is the constant term.

Polynomials Degree

  • Degree of a Polynomial of one Variable

 If a polynomial is in one variable, then the variable with the highest non-negative integral power is called its degree.

Ex- x³+8x²+x+2. Here there are 4 terms namely x³, 8x², x, 2 and the variable with the highest power is x³. So the degree of the above polynomial is 3.

  • Degree of a Polynomial of two or more Variables

 In this case, we add the powers of the variables in each term, and the highest sum obtained is called its degree.

Ex- 8y³+5x²y²+x-2.

Degree of this polynomial is- 4 (2nd Term i.e., 5x²y² = Sum of powers- 2+2)

Various Polynomials Degree

  1. Linear Polynomial – Having degree 1. Ex- x – 2
  2. Quadratic Polynomial- Having degree 2. Ex- y² + x – 2
  3. Cubic Polynomial- Having degree 3. Ex- 8y³ + 5x² + x
  4. Biquadratic Polynomial- Having degree 4. Ex- 8y³ + 5x²y² + x – 2

Constant Polynomial

A polynomial containing only one term, i.e, a constant term is called a constant polynomial. Its degree is zero.

Ex- 2, 1/8 etc. 

Zero Polynomial

A polynomial containing only one term, i.e, zero is called a zero polynomial. Its degree is not defined.

Quadratic Polynomials Factorization

Polynomials Factorization

When a given polynomial (x²-9) is expressed as the product of polynomials (x-3, x+3) each with a lower degree. 

  • Type-1> If polynomial is of the form x²+bx+c

Find integers ‘p’ and ‘q’ such that p+q= b; and pq=c

Now,  x²+bx+c = (x+p)(x+q)

  • Type- 2> If polynomial is of the form ax²+bx+c

Find integers ‘p’ and ‘q’ such that p+q= b; and pq=ac

Now,  x²+bx+c = (ax+p)(ax+q)

Note- Take care of minus(-) and plus(+) signs.

Polynomials Identities for Factorization

  • (a²-b²) = (a+b)(a-b)

Ex- (i) 16x²-25y² (ii) x³-x (iii) 3a³-48

  • (a+b+c) = a² + b² + c² + 2ab + 2bc + 2ca

Ex- (i) 4a² + b² + c² – 4ab – 2bc + 4ca

  • (x + y)³ = x³ + y³ + 3xy(x + y)
  • (x – y)³ = x³ – y³ – 3xy(x – y)

Ex- (i) 100³ (ii) 498³ (iii) (x-5y)³ (iv) (8a+5b)³

  • (x³ + y³) = (x + y)(x² – xy + y²)
  • (x³ – y³) = (x – y)(x² + xy + y²)
  • (x³ + y³ + z³ – 3xyz) = (x + y + z)(x² + y² + z² – xy – yz – zx)
  • If (x + y + z) = 0 ⇒ (x³ + y³ + z³) = 3xyz

Must Read: 40+ Algebraic Identities formula

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