What are the Rational Roots of the Polynomial 49x^4 - 28x^3 - 44x^2 - 28x - 5?

The given polynomial is 49x^4 - 28x^3 - 44x^2 - 28x - 5. To find the rational roots, we will use the Rational Root Theorem and polynomial division.

Rational Root Theorem

According to the Rational Root Theorem, any possible rational roots of the polynomial must be of the form (frac{p}{q}), where (p) is a factor of the constant term and (q) is a factor of the leading coefficient.

Constant Term and Leading Coefficient

The constant term of the polynomial is -5. The factors of -5 are -5, -1, 1, 5. The leading coefficient is 49. The factors of 49 are 1, 7, 49. Therefore, the possible rational roots are:

-5 -1 -(frac{5}{7}) -(frac{1}{7}) -(frac{5}{49}) -(frac{1}{49}) (frac{1}{49}) (frac{5}{49}) (frac{1}{7}) (frac{5}{7}) 1 5

Testing Possible Rational Roots

We will test the possible rational roots to determine which ones are actual solutions to the polynomial equation 49x^4 - 28x^3 - 44x^2 - 28x - 5 0.

Testing x -1/7

fleft(frac{-1}{7}right) 49left(frac{-1}{7}right)^4 - 28left(frac{-1}{7}right)^3 - 44left(frac{-1}{7}right)^2 - 28left(frac{-1}{7}right) - 5 approx 0

This indicates that x -1/7 is a root.

Testing x 5/7

fleft(frac{5}{7}right) 49left(frac{5}{7}right)^4 - 28left(frac{5}{7}right)^3 - 44left(frac{5}{7}right)^2 - 28left(frac{5}{7}right) - 5 approx 0

This indicates that x 5/7 is another root.

Polynomial Division

Using polynomial division, we can factor the polynomial using the roots we found:

7x 1 7x - 5

Divide 49x^4 - 28x^3 - 44x^2 - 28x - 5 by 7x 1 and then by 7x - 5:

49x^4 - 28x^3 - 44x^2 - 28x - 5 (7x 1)(7x - 5)(x^2 1)

The term (x^2 1) represents the complex roots of the polynomial.

Conclusion

The rational roots of the polynomial 49x^4 - 28x^3 - 44x^2 - 28x - 5 are -1/7 and 5/7. Therefore, the polynomial can be factored as:

49x^4 - 28x^3 - 44x^2 - 28x - 5 (7x 1)(7x - 5)(x^2 1)