Operations on Polynomial Class 10th
Introduction
Operations on polynomials are the addition of polynomials, the subtraction of polynomials, the multiplication of polynomials, and the division of polynomials.
➤ Addition of polynomials
➤ Subtraction of polynomials
➤ Multiplication of polynomials
➤ Division of polynomials
Addition of Polynomials
When we add two or more polynomials then only like terms are added. it means terms with the same variables and powers are added. unlike terms will not be added, they will remain unchanged. In addition, the degree of the resultant polynomial remains the same.
Example — Add polynomials 3x^2 + 4xy + 2y^2 and 5y^2 — xy + 7x^2
Solution — 3x^2 + 4xy + 2y^2 + 5y^2 — xy + 7x^2
(3x^2 + 7x^2) + (4xy — xy) + (2y^2 + 5y^2)
10x^2 + 3xy + 7y^2 Ans.
Subtraction of Polynomials
Subtraction of the polynomials is the same as the addition of the polynomials. In this, like terms are subtracted and the unlike terms will be unchanged. In this also, the degree of the resultant polynomial will remain the same.
Example — Subtract the polynomial 3xy^2 + 4xy — 6x^2y + y^3 from 8xy^2 + 7xy + x^2y.
Solution — 8xy^2 + 7xy + x^2y — (3xy^2 + 4xy — 6x^2y + y^3)
8xy^2 + 7xy + x^2y — 3xy^2–4xy + 6x^2y — y^3
(8xy^2–3xy^2) + (7xy — 4xy) + (x^2y + 6x^2y) — y^3
5xy^2 + 3xy + 7x^2y — y^3 Ans.
Multiplication of Polynomials
When two or more polynomials are multiplied then the result is always of a higher degree polynomial. But in two polynomials, if one or both of the polynomials are constant polynomials then the degree will remain the same. In the multiplication of polynomials, powers of the same variables are added by the laws of exponents.
Multiplying a Monomial by a Monomial
Multiplying two monomials
Examples — 1) 4x⨯6y = (4⨯6)⨯(x⨯y) = 24xy
2) 3a⨯(-2b) = {3⨯(-2)}⨯(a⨯b) = -6ab
3) 6p^2⨯8p = (6⨯8)⨯(p^2⨯p) = 48p^3
4) x^4yz^2⨯2xy^3z = (1⨯2)⨯(x^4⨯x⨯y⨯y^3⨯z^2⨯z) = 2x^5y^4z^3 Ans.
Multiplying three or more monomials
Examples — 1) 7t⨯2s⨯3r = (7⨯2⨯3)⨯(t⨯s⨯r) = 42tsr Ans.
2) 4pq⨯5p^2q^2⨯6p^3q^3
= (4⨯5⨯6)⨯(p⨯p^2⨯p^3⨯q⨯q^2⨯q^3)
= 120p^6q^6 Ans.
3) 2x^2y⨯(-4y^2z)⨯(-7z^2x)⨯2x^2yz
= [2x^2y⨯(-4y^2z)]⨯[(-7z^2x)⨯2x^2yz]
= (-8x^2y^3z)⨯(-14x^3yz^3)
= 112x^5y^4z^4 Ans.
Multiplying a Monomial by a Polynomial
Multiplying a monomial by a binomial
Examples — 1) 2a⨯(3b+4)
Using Distributive law,
= 2a⨯3b + 2a⨯4
= 6ab + 8a Ans.
2) 7xy⨯(y^2+5)
= 7xy⨯y^2 + 7xy⨯5
= 7xy^3 + 35xy Ans.
Multiplying a monomial by a trinomial
Examples — 1) 3v⨯(4v^2+5v+7)
Using Distributive law,
= 3v⨯4v^2 + 3v⨯5v + 3v⨯7
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= 12v^3 + 15v^2 + 21v Ans.
2) x^2⨯(9xz+6y^2-xyz)
= x^2⨯9xz + x^2⨯6y^2 — x^2⨯xyz
= 9x^3z + 6x^2y^2 — x^3yz Ans.
Multiplying a Polynomial by a Polynomial
Multiplying a binomial by a binomial
Examples — 1) (2x+3y)⨯(3x+2y)
= 2x⨯(3x+2y) + 3y⨯(3x+2y)
= 2x⨯3x + 2x⨯2y + 3y⨯3x + 3y⨯2y
= 6x^2 + 4xy + 9xy + 6y^2
= 6x^2 + 13xy + 6y^2 Ans.
2) (p^2–6qr)⨯(9q^2+7r)
= p^2⨯(9q^2+7r) — 6qr⨯(9q^2+7r)
= p^2⨯9q^2 + p^2⨯7r — 6qr⨯9q^2–6qr⨯7r
= 9p^2q^2 + 7p^2r — 54q^3r — 42qr^2 Ans.
Multiplying a binomial by a trinomial
Examples — 1) (a+5)⨯(a^2+2a-7)
= a⨯(a^2+2a-7) + 5⨯(a^2+2a-7)
= a⨯a^2+a⨯2a-a⨯7 + 5⨯a^2+5⨯2a-5⨯7
= a^3+2a^2–7a + 5a^2+10a-35
= a^3+7a^2+3a-35 Ans.
2) (x^2–3)⨯(xy+yz+zx)
= x^2⨯(xy+yz+zx) — 3⨯(xy+yz+zx)
= x^2⨯xy+x^2⨯yz+x^2⨯zx — 3⨯xy — 3⨯yz — 3⨯zx
= x^3y + x^2yz + x^3z — 3xy — 3yz — 3zx Ans.
Division of Polynomials
In the division of polynomial, the result is a lesser degree polynomial and if one of the polynomials is a constant polynomial then the degree will remain the same.
Dividing a Monomial by a Monomial
For easy Division, we factorize both Dividend and Divisor in this type of Division.
Examples — 1) 6x^4 ÷ 3x
= 2⨯3⨯x⨯x⨯x⨯x/3⨯x = 2⨯x⨯x⨯x = 2x^3 Ans.
2) 72p^2q^3r ÷ (- 6p^4q)
= 2⨯2⨯2⨯3⨯3⨯p⨯p⨯q⨯q⨯q⨯r/-2⨯3⨯p⨯p⨯p⨯p⨯q
= 2⨯2⨯3⨯q⨯q⨯r/-p⨯p = 12q^2r/-p^2 Ans.
Dividing a Polynomial by a Monomial
Examples — 1) (2xy + 6x) ÷ 2x
= (2xy + 6x)/2x = 2x(y + 3)/2x = (y+3) Ans.
2) (8y^3 + 6y^2 + 12y) ÷ 2y
= (8y^3 + 6y^2 + 12y)/2y = 2y(4y^2 + 3y+ 6)/2y
= 4y^2 + 3y+ 6 Ans.
This can be also solved as = 8y^3/2y + 6y^2/2y + 12y/2y = 4y^2 + 3y+ 6
Dividing a Polynomial by a Polynomial
Examples — 1) (7a^2 + 14a) ÷ (a + 2)
= (7a^2 + 14a)/(a + 2) = 7a(a + 2)/(a + 2) = 7a Ans.
2) (x^4–5x^3–24x^2) ÷ x(x — 8)
= (x^4–5x^3–24x^2)/x(x — 8) = x^2(x^2–5x — 24)/x(x — 8)
= x(x^2–8x + 3x — 24)/(x — 8) [factorization of (x^2–5x — 24)]
= x{x(x — 8) + 3(x — 8)}/(x — 8)
= x(x — 8)(x + 3)/(x — 8) = x(x + 3) Ans.
If the above method does not help you to divide a polynomial by a polynomial, then there is also another method to divide which is called the Division Algorithm.
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