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 $36 x^{3} + 9 x^{2} + 45 x$ , by another polynomial, $9 x^{2}$ . 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 $36 x^{3} + 9 x^{2} + 45 x$ is divided by the quadratic polynomial $9 x^{2}$ . The process involves manipulating the terms of the polynomials to simplify the expression and to find the result in terms of $x$ .

$(36 x^{3} + 9 x^{2} + 45 x) \div 9 x^{2}$

Answer

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Solution:

Step 1:

Express the division as a fraction: $\frac{36 x^{3} + 9 x^{2} + 45 x}{9 x^{2}}$ .

Step 2:

Extract the common factor of $9 x$ from the numerator.

Step 2.1:

Take $9 x$ out of $36 x^{3}$ : $\frac{9 x (4 x^{2}) + 9 x^{2} + 45 x}{9 x^{2}}$ .

Step 2.2:

Take $9 x$ out of $9 x^{2}$ : $\frac{9 x (4 x^{2}) + 9 x (x) + 45 x}{9 x^{2}}$ .

Step 2.3:

Take $9 x$ out of $45 x$ : $\frac{9 x (4 x^{2}) + 9 x (x) + 9 x (5)}{9 x^{2}}$ .

Step 2.4:

Combine the terms with $9 x$ : $\frac{9 x (4 x^{2} + x) + 9 x (5)}{9 x^{2}}$ .

Step 2.5:

Final extraction of $9 x$ : $\frac{9 x (4 x^{2} + x + 5)}{9 x^{2}}$ .

Step 3:

Eliminate the common factor of $9$ .

Step 3.1:

Remove the $9$ : $\frac{9 x (4 x^{2} + x + 5)}{9 x^{2}}$ .

Step 3.2:

Simplify the expression: $\frac{x (4 x^{2} + x + 5)}{x^{2}}$ .

Step 4:

Cancel out the common $x$ factors.

Step 4.1:

Factor $x$ from $x^{2}$ : $\frac{x (4 x^{2} + x + 5)}{x \cdot x}$ .

Step 4.2:

Remove the $x$ : $\frac{x (4 x^{2} + x + 5)}{x \cdot x}$ .

Step 4.3:

Simplify further: $\frac{4 x^{2} + x + 5}{x}$ .

Step 5:

Divide the terms individually by $x$ : $\frac{4 x^{2}}{x} + \frac{x}{x} + \frac{5}{x}$ .

Step 6:

Simplify each term by canceling $x$ .

Step 6.1:

Cancel $x$ in $4 x^{2}$ : $\frac{4 x}{1} + \frac{x}{x} + \frac{5}{x}$ .

Step 6.2:

Simplify the fractions: $4 x + 1 + \frac{5}{x}$ .

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.