Polynomials Definition and Concepts
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.
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.
Symbols having fixed numerical values are called constants.
For Example- 2, 3, 1/2, 22/7, etc.
When we combine the variables and constants along with operations +, -, ×, and ÷ (few or all), the result obtained is called an algebraic expression.
Note- This above expression contains 3 terms (Home Work- Name these)
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.
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.
Degree of this polynomial is- 4 (2nd Term i.e., 5x²y² = Sum of powers- 2+2)
Various Polynomials Degree
- Linear Polynomial – Having degree 1. Ex- x – 2
- Quadratic Polynomial- Having degree 2. Ex- y² + x – 2
- Cubic Polynomial- Having degree 3. Ex- 8y³ + 5x² + x
- Biquadratic Polynomial- Having degree 4. Ex- 8y³ + 5x²y² + x – 2
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.
A polynomial containing only one term, i.e, zero is called a zero polynomial. Its degree is not defined.
Quadratic 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