Division of Algebraic Expressions

Division is the inverse operation of multiplication.

1. Process to divide a monomial by another monomial

  • Write the irreducible factors of both the monomials
  • Cancel out the common factors.
  • The balance is the answer to the division.

Example

Solve \mathrm{54y^3}\div9y

Solution:

Write the irreducible factors of the monomials

\mathrm{54y^3 = 3 \times 3 \times 3 \times 2 \times y \times y \times y}

 \mathrm{9y = 3 \times 3 \times y}

 \mathrm{\dfrac{54y^3}{9y}=\dfrac{3\times3\times3\times2\times y\times y \times y}{3\times3\times y}=2\times3\times y\times y=6y^2}

2. Process to divide a polynomial by a monomial

  • Write the irreducible form of the polynomial and monomial both.
  • Take out the common factor from the polynomial.
  • Cancel out the common factor if possible.
  • The balance will be the required answer.

Example

Solve \mathrm{4x^3+2x^2+2x\div2x} .

Solution:

Write the irreducible form of all the terms of polynomial

\mathrm{4x^3+2x^2+2x}

\mathrm{= 4(x)\ (x)\ (x) + 2(x)\ (x) + 2x}

Take out the common factor i.e. 2x

\mathrm{= 2x\ (2x^2 + x + 1)}

\mathrm{\dfrac{4x^3+2x^2+2x}{2x}=\dfrac{2x(2x^2+x+1)}{2x}=(2x^2+x+1)}

3. Process to divide a polynomial by a polynomial

In the case of polynomials we need to reduce them and find their factors by using identities or by finding common terms or any other form of factorization. Then cancel out the common factors and the remainder will be the required answer.

Example

Solve \mathrm{z\ (5z^2-80)\div5z\ (z+4)}

Solution:

Find the factors of the polynomial

\mathrm{=z\ (5z^2-80)}

\mathrm{= z [(5 \times z^2) - (5 \times 16)]}

\mathrm{= z \times 5 \times (z^2 - 16)}

\mathrm{= 5z \times (z + 4)\ (z - 4)} [using the identity \mathrm{a^2 - b^2 = (a + b)\ (a - b)} ]

\mathrm{\dfrac{\left\{z(5z^2-80)\right\}}{5z(z+4)}=\dfrac{5z(z+4)(z-4)}{5z(z+4)}=z-4}

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