HCF AND LCM OF POLYNOMIALS-TIPS

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IMPORTANT TIPS

• Make sure that you know how to find HCF(Highest Common Factor) and LCM(Lowest Common Multiple) of given set of numbers.



• p(x)=a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + ... + a₁x+a₀ is called the general form of a polynomial in x, where a_i∈ R(i=0,1,2,3,...) and a_n≠ 0.



• n is called the degree of polynomial.



• p(x) is the symbol for a polynomial over x and q(y) is the symbol for a polynomial over y.Similarly, we use r(x), m(x), n(y), p(t), s(t), etc. to denote polynomials symbolically.



• If p(x)= 0, then it is called the ZERO POLYNOMIAL.



• If p(x)= k, (k∈ R, a constant), then it is called a CONSTANT POLYNOMIAL.



• If n = 1, the polynomial is called a LINEAR POLYNOMIAL. Thus, p(x)= 3x + 2 is a linear polynomial.



• If n = 2, the polynomial is called a SECOND-DEGREE POLYNOMIAL or a QUADRATIC POLYNOMIAL. Thus, p(x)=2x² - 3x - 3 is a second-degree polynomial.



• The highest exponent(index) of the variable denotes the degree of a polynomial.Thus, m(y)= 6y⁵ - 4y³ + x² - 3x + 1 is a polynomial of degree 5.



• The standard method of writing a polynomial is to write it either in the ascending order or the descending order of the exponent of its variable.Thus, p(x)= x⁴ - 3x³ + 2x² + 5x + 6 (descending order) or p(x)= 6 + 5x + 2x² - 3x³ + x⁴ (ascending order).



• In the general form of a polynomial, a_i is the coefficient of xⁱ (i=1,2,3,...). Thus, for i = 0, we have a₀x⁰ = a₀ ( because x⁰ = 1). ∴ a₀ is called the CONSTANT TERM.



• If a polynomial p(x) is the product of polynomials g(x) and h(x), then g(x) and h(x) are aclled the factors of p(x).



• HCF (GCF) OF POLYNOMIALS:If h(x) is a common factor of the given set of polynomials and every common factor of the given polynomials is a factor of h(x), then h(x) is said to be the Highest(Greatest) Common Factor of the given set of polynomials.



• LCM OF POLYNOMIALS:If m(x) is a common multiple of the given set of polynomials and every common multiple of given polynomials is also a multiple of m(x), then m(x) is said to be the Least Common Multiple of the given set of polynomials.



• If h(x) and m(x) are the HCF and LCM, respectively, of two polynomials p(x) and q(x), then
  
  

  p(x) · q(x) = ± h(x) · m(x)

  
  


• ALERT:
  
  

  p(x) · q(x) · r(x) ≠ ± h(x) · m(x)
  

  
  
  
  The relation holds only for two polynomials.