Division of a Polynomial by a Monomial The division of a polynomial by a monomial is a process that involves breaking down the larger polynomial into smalle...
Division of a Polynomial by a Monomial
The division of a polynomial by a monomial is a process that involves breaking down the larger polynomial into smaller parts and combining them to form the quotient polynomial. This technique helps us simplify and evaluate polynomial expressions by isolating the variable terms in the quotient.
Step 1: Factor the Polynomial
Begin by factoring the original polynomial into two or more linear factors using the factorisation theorem. A linear factor is a polynomial of the form ax + b, where a and b are constants.
Step 2: Divide the Factored Polynomials
Divide each factor by the monomial using long division or synthetic division. This process involves repeatedly dividing the leading coefficient by the corresponding degree of the monomial and adding the resulting coefficients to the quotient.
Step 3: Combine Like Terms
Arrange the quotient polynomial in descending order of degree, with the highest degree term on top. Collect like terms by combining polynomials with the same variable exponents.
Step 4: Write the Quotient Polynomial
The quotient polynomial represents the result of dividing the original polynomial by the monomial. It is a polynomial that represents the result of the division.
Example:
Division of x^2 + 3x + 2 by x + 1:
Factorization: x^2 + 3x + 2 = (x + 2)(x + 1)
Division: (x + 2)(x + 1) / (x + 1) = x + 2
Therefore, the quotient polynomial is x + 2.