This blog post is going to tell you about polynomials , and if you want to learn them , then believe me that this is the right place for you .
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An algebraic expression having a variable x and its non-negative integral powers with real numbers as coefficients is called a Polynomial in x.
DEGREE OF A POLYNOMIAL
The exponent of the highest degree term in a polynomial is known as degree of polynomial.
1. 4x³ - 9x² +7x +9 is a polynomial in variable x of degree 3.
2. 3x² + 9x -1 +7/x is not a polynomial . As it contains a term namely 7/x having negative integral powers of variable x.
3. 5x² - 7x⁷ⁱ² + x -1 is not a polynomial as the term - 7x⁷ⁱ² contains rational power of variable of x.
Okay champ you are doing good !
Lets learn some more concepts .
VARIOUS TYPES OF POLYNOMIALS
1. BASED ON DEGREE OF POLYNOMIAL
(a) Zero Polynomial : A polynomial of degree zero is called zero polynomial i.e. F(x) = 8.
The degree of zero polynomial is not defined.
(b) Linear polynomial : A polynomial of degree 1 is called a linear polynomial .
e.g. 9x+7 , x-9 , etc
(c) Quadratic polynomial : A polynomial with degree 2, is called a quadratic polynomial. The general form of quadratic polynomial is ax²+bx+c, where a,b and c are real and a≄ 0 .
(d)Cubic polynomial: A polynomial of degree 3, is called a cubic polynomial. The general form of a cubic polynomial is ax³+bx²+cx+d, where a,b,c and d are real and a≄0.
(e) Biquadratic polynomial: Polynomial of degree 4 is called a biquadratic polynomial.
2. BASED ON NUMBER OF TERMS
(a) Monomial : A polynomial containing only one term is called a monomial e.g. 6x².
(b) Binomial : A polynomial containing two terms is called a binomial e.g. 5x³+7x
(c) Trinomial : A polynomial containing three terms is called a trinomial e.g. 7x²+5x+6
FACTORS AND MULTIPLES
If a number divides another number exactly then the number which divides , is called a factor of the number that has been divided is known as the multiple of the number which divides it .
e.g. x and 2x²+x+1 are factors of 2x³+x²+x and 2x³+x²+x is a multiple of x as well as multiple of 2x²+x+1.
To express a polynomial as the product of other polynomials of degree less than that of the given polynomial is called as factorisation.
e.g x² - 49 = x² - 7² = (x - 7)(x + 7)
REQUIRED FORMULAE FOR FACTORISATION* (a² - b²) = (a + b)(a - b)
* (a + b)² = a² + b² + 2ab and (a - b)² = a² + b² -2ab
* (a + b)² - (a - b)² = 4ab and (a + b)² + (a - b)² = 2 (a² + b²)
* (a + b)³ = a³ + b³ + 3ab(a + b)
* (a - b)³ = a³ -b³ - 3ab(a - b)
* (a³ + b³) = (a - b)(a² + b² - ab)
* (a³ - b³) = (a - b)(a² + b² + ab)
*(a + b + c)² = (a² + b² + c² + 2(ab + bc + ac))
*(a³ + b³ + c³ - 3abc) = (a + b +c)(a² + b² + c² - ab - ac - bc)
* if a + b + c = 0 => a³+ b³+ c³ = 3abc
*a⁴+ a²b²+ b⁴ = (a² + ab + b²)(a² - ab + b²)
METHODS OF FACTORISATION
1. Factorisation by taking out the common factor
If each term of an expression has a common factor, take out the common factors and then divide each term by this common factor.
factorise value of 16x² + 12xy
16x² + 12xy =4.4x² + 3.(4xy) = 4x(4x + 3y)
2. Factorisation by grouping
Sometimes in a given polynomial , it is not possible to take out a common factor directly. However on rearranging the terms of the polynomial and grouping such that all the terms have a common factor.
Factorise value of xy + yz + xa + za
xy + yz+ xa + za = y(x + z) + a (x + z)= (x + z )(y + a)
3. Factorisation by perfect square polynomials
Here the polynomial is given as the perfect square quadratic polynomial, which is the square of a binomial.
(a + b)² = a² + 2 ab + b²
(a - b)² = a² - 2ab + b²
4. Factorising the difference of two squares
In case the polynomial is given in the form of a²- b² to evaluate it following formulae is applied.
a² - b² = (a - b)(a + b)
5. Factorisation of quadratic polynomials
quadratic polynomial of the type ax² + bx + c and pq =c , then
ax² + bx + c = ax² + (p + q)x + pq = ax² + px + qx + pq
6. Factorisation of sum and difference of cubes
In case , the polynomial is given in the form of a³ + b³ or a³ - b³ to evaluate it following formula is applied.
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
7. Factorisation of a³ + b³ + c³ - 3abc
Here it is easy to use
a³+ b³ + c³ -3abc = (a + b + c )(a² + b² + c² - ab - bc - ac)
If a + b + c =0 then a³ + b³ + c³ = 3abc
Long BLOG POST ?
No, i think informative one!
If p(x) and g(x) any two polynomials with g(x)≄ 0, then we can find polynomials q(x) and r (x) such that
p(x) = g(x) + q(x) + r (x)
i.e. dividend = (divisor * quotient )+ remainder
Lets do some exercises together , buddy!
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