Division of a Polynomial by a Monomial
In this section, we delve into the process of dividing a polynomial by a monomial, an essential skill in algebra. A polynomial can consist of multiple terms, and when we want to divide such an expression by a monomial, we can utilize two effective methods.
Key Concepts:
- Factorization: The polynomial is expressed as a sum of terms, each of which can be factorized to identify common factors with the monomial.
- Common Factors: We take the common factor out from each term in the polynomial, allowing us to simplify the expression.
- Division of Each Term: Alternatively, we can directly divide each term of the polynomial by the monomial, which often leads to a straightforward simplification.
Example:
For instance, consider dividing the polynomial 4y³ + 5y² + 6y by the monomial 2y. We can express each term in this polynomial in terms of 2y, allowing for simplification:
- Factor out
2y:
4y³ + 5y² + 6y = 2y(2y² + y + 3)
The division then yields a simplified result as follows:
(4y³ + 5y² + 6y) ÷ 2y = 2y² + y + 3.
Through this section, students will understand how to effectively handle polynomial expressions when divided by monomials, vital for more advanced algebraic concepts.
Similar Question
Example : Divide \( 18(x^2y + y^2z + z^2x) \) by \( 6xyz \) using both methods.
Solution:
\[ 18(x^2y + y^2z + z^2x) = 18 \cdot (x^2y) + 18 \cdot (y^2z) + 18 \cdot (z^2x) \]
\[ = 3 \cdot 6(x^2y + y^2z + z^2x) \]
Taking out the common factor, we have:
\[ = 3\cdot6(x + y + z) \]
Therefore,
\[ \frac{18(x^2y + y^2z + z^2x)}{6xyz} = 3(x + y + z) \]
Alternately, \( 18(x^2y + y^2z + z^2x) \) can also be simplified to \( 3x^2y + 3y^2z + 3z^2x = 3(x^2y + y^2z + z^2x) \)
\[ = 3\cdot 6(x + y + z) \]