To factor the polynomial $15x^4 + 91x^3 + 79x^2 - 7x - 10$ completely using the Rational Root Theorem, let's proceed with the following steps.

Step 1: Identify Potential Rational Roots

The Rational Root Theorem states that any rational root, $\frac{p}{q}$, of the polynomial must have $p$ as a factor of the constant term (-10) and $q$ as a factor of the leading coefficient (15).

Factors of -10 (constant term):

$\pm 1, \pm 2, \pm 5, \pm 10$

Factors of 15 (leading coefficient):

$\pm 1, \pm 3, \pm 5, \pm 15$

Possible Rational Roots:

Using the factors above, the possible rational roots are: $\pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{5}{3}, \pm \frac{10}{3}, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{5}{5} (= \pm 1), \pm \frac{10}{5} (= \pm 2), \pm \frac{1}{15}, \pm \frac{2}{15}, \pm \frac{5}{15} (= \pm \frac{1}{3}), \pm \frac{10}{15} (= \pm \frac{2}{3})$

This gives us a set of possible rational roots: $\pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{5}{3}, \pm \frac{10}{3}, \pm \frac{1}{5}, \pm \frac{2}{5}$

Step 2: Test Each Potential Root

Using synthetic division or substitution, we test each potential root to see if it yields a remainder of zero.

After testing, we find that $x = -\frac{1}{3}$ is a root of the polynomial.

Step 3: Factor Out $(3x + 1)$

Since $x = -\frac{1}{3}$ is a root, $(3x + 1)$ is a factor of the polynomial. We can use synthetic division to divide the polynomial by $3x + 1$.

Synthetic Division

Dividing $15x^4 + 91x^3 + 79x^2 - 7x - 10$ by $3x + 1$, we obtain: $15x^4 + 91x^3 + 79x^2 - 7x - 10 = (3x + 1)(5x^3 + 28x^2 + 17x - 10)$

Step 4: Factor the Quotient $5x^3 + 28x^2 + 17x - 10$

We repeat the Rational Root Theorem for $5x^3 + 28x^2 + 17x - 10$. Testing possible rational roots, we find that $x = 2$ is a root.

Factor Out $(x - 2)$

Since $x = 2$ is a root, $(x - 2)$ is a factor. Dividing $5x^3 + 28x^2 + 17x - 10$ by $x - 2$, we obtain: $5x^3 + 28x^2 + 17x - 10 = (x - 2)(5x^2 + 38x + 5)$

Step 5: Factor $5x^2 + 38x + 5$

Now, we factor $5x^2 + 38x + 5$ by looking for factors of 5 and 5 whose product equals 25 and sum equals 38.

The quadratic factors as: $5x^2 + 38x + 5 = (5x + 1)(x + 5)$

Final Factorization

Combining all the factors, we get the complete factorization: $15x^4 + 91x^3 + 79x^2 - 7x - 10 = (3x + 1)(x - 2)(5x + 1)(x + 5)$

Would you like more details on any step?

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1. How do you use synthetic division to confirm a root?
2. Can you explain the Rational Root Theorem in simpler terms?
3. How do you approach factoring a cubic polynomial after finding one factor?
4. What techniques are used to factor quadratics?
5. How do you determine if a polynomial is fully factored?

Tip: Always verify each factor by expanding back to the original polynomial to avoid errors.