Division of algebraic expressions is a fundamental skill in mathematics. By mastering the technique of factorisation, you can simplify complex expressions and solve problems with ease. This guide breaks down the division process for both monomials and polynomials.
A monomial is an algebraic expression with only one term. When dividing one monomial by another, the key is to break down both expressions into their prime factors (coefficients and variables) and then cancel out the common factors.
1) Divide $6x^3$ by $2x$
Solution:
Cancel the common factors ($2$ and $x$):
2) Divide $-20x^4$ by $10x^3$
Solution:
Cancel the common factors ($10$ and three $x$’s):
3) Divide $7x^2y^2z^2$ by $14xyz$
Solution:
Cancel the common factors ($7$, $x$, $y$, and $z$):
A polynomial is an algebraic expression with one or more terms (a sum of several monomials). When dividing a polynomial by a monomial, we use factorisation to simplify the numerator (the polynomial) and then cancel out any common factors with the denominator (the monomial).
1) Divide $24(x^2yz + xy^2z + xyz^2)$ by $8xyz$
Solution:
First, write the expression as a fraction:
Next, factorise the polynomial in the numerator by taking out the greatest common factor ($xyz$) from the terms inside the parentheses:
Now, cancel the common factors in the numerator and the denominator ($xyz$ and the coefficients $24/8$):
2) Divide $3y^8 – 4y^6 + 5y^4$ by $y^4$
Solution:
Write the expression as a fraction:
Method A: Splitting the Terms (as done in the image example)
Factor out the smallest power of $y$, which is $y^4$, from the numerator:
Cancel the common factor $y^4$:
(Alternatively, you could divide each term in the polynomial separately by the monomial: $\frac{3y^8}{y^4} – \frac{4y^6}{y^4} + \frac{5y^4}{y^4} = 3y^{8-4} – 4y^{6-4} + 5y^{4-4} = 3y^4 – 4y^2 + 5$)