Today is our topic of discussion Multiplication and Division of Polynomial .
Multiplication and Division of Polynomial
Addition, Subtraction and Multiplication of two polynomial is always polynomial, but division of polynomial may or may not be polynomial. For example, if we divide x by x³, take the answer as x-2, then it is not a polynomial but if we take r as a remainder and result as 0, it is a polynomial.
Division Algorithm
If both P(x) and Q(x) are the polynomials of the variable x and if degree of Q(x) ≤ degree of P(x), we can divide P(x) by Q(x) in usual manner and we obtain a quotient F(x) and a remainder R(x), where,
1) Both F(x) and R(x) are the polynomial of the variable x
2) degree of F(x) = degree of P(x)- degree of Q(x)
3) either R(x) = 0 or degree of R(x) < degree of Q(x)
4) P(x) = F(x)Q(x) + R(x) holds for all x.
Equality Rule of Polynomials
1) If ax+b=px+q holds for all x, putting x = 0 and x = = 1 we get respectively, b = q and a + b = p+q, from whence it is found a = p, b =q
2) If ax² + bx + c = px²+qx+r holds for x, putting x = 0, x = 1 and x = -1 we get respectively c=r, a+b+c=p+q+r and a-b+c=p-q+r from whence it is found that a = p, b=q, c=r.
3) In general, if aqx*n + a₁x²-¹ + ɑ²x²-² + ··· + An-1X + a₂ = Pоx² + P₁x²-1 + -2 Pn-1, an Pn. +Pn-12+ Pn holds for all x, ao Po, a1 = P1, an-1 =
i.e., the two coefficients of a with the same power are equal in both sides of an equal sign.
Remarks: It is convenient to denote the coefficients of a polynomial of degree n by ao(a sub-zero), a₁ (a sub- one) etc.
Identity
If the two polynomials P(x), Q(x) for all x are equal, their equality is called the referred to as identity of the polynomials; sometimes this is indicated by writing P(x) = Q(x). In this case the two polynomials P(r) and Q(x) are the same. The sign is called the identity sign. Generally, the equality of the two algebraic expressions is called the identity, if the domain of any variable of the two expressions is the same and for the values included in the domain of the variables, the values of the two expressions are equal. For example, x(x + 2) = x² + 2x, (x + y)² = x² + 2xy + y² are identities.
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