Simplify (75b^5-36b^4+18b^3)/(-6b^3)

Solution:

Step 1: Extract the common factor from the numerator

Extract $3b^3$ from each term in the numerator $75b^5-36b^4+18b^3$.

Step 1.1

Take $3b^3$ out of $75b^5$: $\frac{3b^3(25b^2)-36b^4+18b^3}{-6b^3}$

Step 1.2

Take $3b^3$ out of $-36b^4$: $\frac{3b^3(25b^2)+3b^3(-12b)+18b^3}{-6b^3}$

Step 1.3

Take $3b^3$ out of $18b^3$: $\frac{3b^3(25b^2)+3b^3(-12b)+3b^3(6)}{-6b^3}$

Step 1.4

Combine the factored terms: $\frac{3b^3(25b^2-12b)+3b^3(6)}{-6b^3}$

Step 1.5

Complete the factoring: $\frac{3b^3(25b^2-12b+6)}{-6b^3}$

Step 2: Simplify the common factors between numerator and denominator

Step 2.1

Factor out the common $3$ in the numerator: $\frac{3(b^3(25b^2-12b+6))}{-6b^3}$

Step 2.2

Eliminate the common factors.

Step 2.2.1

Factor out the $3$ in the denominator: $\frac{3(b^3(25b^2-12b+6))}{3(-2b^3)}$

Step 2.2.2

Cancel out the common $3$: $\frac{\cancel{3}(b^3(25b^2-12b+6))}{\cancel{3}(-2b^3)}$

Step 2.2.3

Simplify the expression: $\frac{b^3(25b^2-12b+6)}{-2b^3}$

Step 3: Cancel the common $b^3$ factor

Step 3.1

Cancel out $b^3$: $\frac{\cancel{b^3}(25b^2-12b+6)}{-2\cancel{b^3}}$

Step 3.2

Rewrite the simplified expression: $\frac{25b^2-12b+6}{-2}$

Step 4: Adjust the negative sign

Place the negative sign in front of the fraction: $-\frac{25b^2-12b+6}{2}$

Knowledge Notes:

The problem involves simplifying a rational expression, which is a fraction where both the numerator and the denominator are polynomials. The steps taken in the solution involve factoring common terms, canceling common factors, and simplifying the expression.

1. Factoring: The process of writing an expression as a product of its factors. In this case, we factored out $3b^3$ from each term in the numerator.

2. Common Factors: These are factors that are the same in both the numerator and the denominator. They can be canceled out to simplify the expression.

3. Simplifying Rational Expressions: After factoring and canceling common factors, the expression is rewritten in its simplest form.

4. Negative Signs: When simplifying expressions, the negative sign can be moved in front of the fraction for clarity.

The use of Latex in the solution helps to clearly display the mathematical expressions and the steps taken to simplify them.