Simplify (75b^5-36b^4+18b^3)/(-6b^3)
The problem asks to simplify a polynomial expression which is in the form of a fraction. The numerator of the fraction consists of a polynomial with three terms, all of which are multiples of powers of a variable b. The denominator is a monomial, a single term also involving a power of b with a negative coefficient. Simplifying the expression involves dividing each term in the numerator by the term in the denominator, reducing the expression to its simplest form, and taking into account the rules for division of powers and handling of negative signs.
$\frac{75 b^{5} - 36 b^{4} + 18 b^{3}}{- 6 b^{3}}$
Answer
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.
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.
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.
Simplifying Rational Expressions: After factoring and canceling common factors, the expression is rewritten in its simplest form.
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.
