Final answer to the problem
Step-by-step Solution
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- Differential
- Find the derivative
- Find the integral
- Find the derivative using the definition
- Solve by quadratic formula (general formula)
- Simplify
- Find the integral
- Find the derivative
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We can factor the polynomial $\left(2x^3+3x+5\right)$ using the rational root theorem, which guarantees that for a polynomial of the form $a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$ there is a rational root of the form $\pm\frac{p}{q}$, where $p$ belongs to the divisors of the constant term $a_0$, and $q$ belongs to the divisors of the leading coefficient $a_n$. List all divisors $p$ of the constant term $a_0$, which equals $5$
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$1, 5$
Learn how to solve rational equations problems step by step online. Solve the rational equation y=((3x+5)^2(2x^3+3x+5))/((x+1)^4). We can factor the polynomial \left(2x^3+3x+5\right) using the rational root theorem, which guarantees that for a polynomial of the form a_nx^n+a_{n-1}x^{n-1}+\dots+a_0 there is a rational root of the form \pm\frac{p}{q}, where p belongs to the divisors of the constant term a_0, and q belongs to the divisors of the leading coefficient a_n. List all divisors p of the constant term a_0, which equals 5. Next, list all divisors of the leading coefficient a_n, which equals 2. The possible roots \pm\frac{p}{q} of the polynomial \left(2x^3+3x+5\right) will then be. Trying all possible roots, we found that -1 is a root of the polynomial. When we evaluate it in the polynomial, it gives us 0 as a result.
