Summary
Highlights
The video introduces the concept of finding possible rational roots of a polynomial using the Rational Root Theorem. It outlines four steps: first, find divisors of the constant term; second, find divisors of the leading coefficient; third, form all possible rational roots from these divisors; and fourth, use the Factor Theorem to determine which of these possible roots are actual roots, noting that not all possibilities will be solutions.
The video then applies the Rational Root Theorem to the function x^4 - 10x^2 - 9. The first step is to find the divisors of the constant term, which is 9 (divisors are 1, 3, 9). The second step is to find the divisors of the leading coefficient, which is 1 (divisors are 1).
The third step involves forming all possible rational roots by dividing the constant term's divisors by the leading coefficient's divisors. This results in possible roots of ±1/1 (±1), ±3/1 (±3), and ±9/1 (±9).
In the fourth step, the Factor Theorem is used to test each possible root. A number is a root if, when substituted into the polynomial, the result is 0. The video demonstrates that substituting 1, -1, 3, and -3 into the function yields 0, confirming they are roots. However, substituting 9 and -9 results in 5760, indicating they are not roots.
Since the highest exponent of the polynomial x^4 - 10x^2 - 9 is 4, there can be up to four roots. The identified roots are 1, -1, 3, and -3. The polynomial can then be expressed in factored form as (x-1)(x+1)(x-3)(x+3).