Simplify (48x^11-12x^10+30x^8+12x^6+5x^5)/(6x^8)
The given problem is a mathematical expression that requires simplification. Specifically, it is asking for the division of a polynomial by a monomial. The polynomial is a sum of multiple terms, each consisting of a coefficient and a variable 'x' raised to a power (the exponent). The monomial is a single term with a coefficient of 6 and the variable 'x' raised to the power of 8. To simplify the expression, you would divide each term of the polynomial by the monomial, essentially reducing the exponents of 'x' by 8 wherever possible and dividing the coefficients by 6. The simplification process should yield a new polynomial, which is the simplified form of the original expression.
$\frac{48 x^{11} - 12 x^{10} + 30 x^{8} + 12 x^{6} + 5 x^{5}}{6 x^{8}}$
Answer
Solution:
Step 1: Extract the common factor
Extract $x^5$ from each term in the numerator $48x^{11} - 12x^{10} + 30x^8 + 12x^6 + 5x^5$.
Step 1.1: Factor out from the first term
Extract $x^5$ from $48x^{11}$ to get $\frac{x^5(48x^6) - 12x^{10} + 30x^8 + 12x^6 + 5x^5}{6x^8}$.
Step 1.2: Factor out from the second term
Extract $x^5$ from $-12x^{10}$ to get $\frac{x^5(48x^6) + x^5(-12x^5) + 30x^8 + 12x^6 + 5x^5}{6x^8}$.
Step 1.3: Factor out from the third term
Extract $x^5$ from $30x^8$ to get $\frac{x^5(48x^6) + x^5(-12x^5) + x^5(30x^3) + 12x^6 + 5x^5}{6x^8}$.
Step 1.4: Factor out from the fourth term
Extract $x^5$ from $12x^6$ to get $\frac{x^5(48x^6) + x^5(-12x^5) + x^5(30x^3) + x^5(12x) + 5x^5}{6x^8}$.
Step 1.5: Factor out from the fifth term
Extract $x^5$ from $5x^5$ to get $\frac{x^5(48x^6) + x^5(-12x^5) + x^5(30x^3) + x^5(12x) + x^5 \cdot 5}{6x^8}$.
Step 1.6: Combine the factored terms
Combine the terms factored by $x^5$ to get $\frac{x^5(48x^6 - 12x^5 + 30x^3 + 12x + 5)}{6x^8}$.
Step 2: Simplify by canceling common factors
Step 2.1: Factor out from the denominator
Factor $x^5$ out of $6x^8$ to get $\frac{x^5(48x^6 - 12x^5 + 30x^3 + 12x + 5)}{x^5(6x^3)}$.
Step 2.2: Cancel the common factor
Cancel out the common $x^5$ factor to get $\frac{\cancel{x^5}(48x^6 - 12x^5 + 30x^3 + 12x + 5)}{\cancel{x^5}(6x^3)}$.
Step 2.3: Rewrite the simplified expression
Rewrite the expression to get $\frac{48x^6 - 12x^5 + 30x^3 + 12x + 5}{6x^3}$.
Knowledge Notes:
Factoring: The process of breaking down an expression into products of other expressions or factors. In this case, we factored out $x^5$ from each term in the numerator.
Common Factor: A term that is present in all parts of an expression. Here, $x^5$ is a common factor in the numerator and $x^5$ is also in the denominator.
Simplifying Fractions: The process of reducing fractions to their simplest form by canceling out common factors in the numerator and denominator.
Polynomial Division: When dividing polynomials, one can cancel out common factors, which is a form of simplifying the expression.
LaTeX Formatting: A typesetting system used to create formatted mathematical expressions. In this solution, LaTeX is used to properly display the mathematical expressions and operations.
