Divide (36x^3+9x^2+45x)÷9x^2
The given problem is a division question involving algebraic expressions. It asks to perform the division of a polynomial, which is in the form of $36x^3 + 9x^2 + 45x$, by another polynomial, $9x^2$. This operation is similar to long division with numbers but applied to polynomial expressions. The objective is to determine the quotient when the cubic polynomial $36x^3 + 9x^2 + 45x$is divided by the quadratic polynomial $9x^2$. The process involves manipulating the terms of the polynomials to simplify the expression and to find the result in terms of $x$.
$\left(\right. 36 x^{3} + 9 x^{2} + 45 x \left.\right) \div 9 x^{2}$
Answer
Solution:
Step 1:
Express the division as a fraction: $\frac{36x^3 + 9x^2 + 45x}{9x^2}$.
Step 2:
Extract the common factor of $9x$ from the numerator.
Step 2.1:
Take $9x$ out of $36x^3$: $\frac{9x(4x^2) + 9x^2 + 45x}{9x^2}$.
Step 2.2:
Take $9x$ out of $9x^2$: $\frac{9x(4x^2) + 9x(x) + 45x}{9x^2}$.
Step 2.3:
Take $9x$ out of $45x$: $\frac{9x(4x^2) + 9x(x) + 9x(5)}{9x^2}$.
Step 2.4:
Combine the terms with $9x$: $\frac{9x(4x^2 + x) + 9x(5)}{9x^2}$.
Step 2.5:
Final extraction of $9x$: $\frac{9x(4x^2 + x + 5)}{9x^2}$.
Step 3:
Eliminate the common factor of $9$.
Step 3.1:
Remove the $9$: $\frac{\cancel{9}x(4x^2 + x + 5)}{\cancel{9}x^2}$.
Step 3.2:
Simplify the expression: $\frac{x(4x^2 + x + 5)}{x^2}$.
Step 4:
Cancel out the common $x$ factors.
Step 4.1:
Factor $x$ from $x^2$: $\frac{x(4x^2 + x + 5)}{x \cdot x}$.
Step 4.2:
Remove the $x$: $\frac{\cancel{x}(4x^2 + x + 5)}{\cancel{x} \cdot x}$.
Step 4.3:
Simplify further: $\frac{4x^2 + x + 5}{x}$.
Step 5:
Divide the terms individually by $x$: $\frac{4x^2}{x} + \frac{x}{x} + \frac{5}{x}$.
Step 6:
Simplify each term by canceling $x$.
Step 6.1:
Cancel $x$ in $4x^2$: $\frac{4x}{1} + \frac{x}{x} + \frac{5}{x}$.
Step 6.2:
Simplify the fractions: $4x + 1 + \frac{5}{x}$.
Knowledge Notes:
Division as a Fraction: Dividing by a term can be represented as multiplying by its reciprocal, hence the division of a polynomial by a monomial is expressed as a fraction.
Factoring: This involves taking out a common factor from each term in a polynomial. It simplifies the expression and is a crucial step in simplifying fractions.
Canceling Common Factors: When a factor appears in both the numerator and denominator, it can be canceled out, as it is equivalent to dividing by that factor.
Simplifying Expressions: After canceling common factors, the expression should be rewritten in its simplest form.
Splitting Fractions: A single fraction with multiple terms in the numerator can be split into multiple fractions, each with its own numerator and a common denominator.
Polynomial Division: When dividing polynomials, each term in the numerator should be divided by the denominator, and common factors should be canceled out to simplify the result.
