91Ó°ÊÓ

When facing a polynomial function such as \(f(x)=2x^3-5x^2+x+2\), the Rational Root Theorem is an essential tool for identifying possible roots. According to this theorem, any rational root of a polynomial with integer coefficients is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.

In our example, the constant term is 2 and the leading coefficient is also 2, so the factor set for both is the same: \(\{1, -1, 2, -2\}\). The possible rational zeros are therefore \(\pm1\) and \(\pm2\). The Rational Root Theorem doesn't tell us which, if any, are actual zeros — it only gives us a list of candidates to test, significantly narrowing down our options.

Once we have a list of potential rational zeros, synthetic division is the next step to determine actual zeros. This process simplifies the calculation by providing a quick method for dividing a polynomial by a binomial of the form \((x-\alpha)\). It's much like long division but streamlined.

Here's how you use it for our exercise: Write down the potential zero, then the coefficients of the polynomial (2, -5, 1, 2). By following the steps of synthetic division (multiply, add, and repeat) for each candidate zero, we identify which give a remainder of zero. In this case, \(-2\) is confirmed as a zero of the polynomial \(f(x)\) because the remainder after synthetic division is zero. Essentially, if synthetic division ends with a remainder of zero, the candidate zero is indeed a root.

The Factor Theorem works hand in hand with synthetic division. It states that if a polynomial \(f(x)\) has a root at \(x=\alpha\), then it can be factored by \((x-\alpha)\). This theorem allows us to reduce the polynomial to a lower degree polynomial once we've identified a root.

After using synthetic division and finding that \(-2\) is a root, we can factor the polynomial by \((x+2)\). This leaves us with a quadratic expression, which is easier to solve. The result of the synthetic division gives us a new, simpler equation where the roots are easier to find, leading us to uncover all zeros of the original polynomial.

Once we've got a quadratic equation after applying the Factor Theorem, we reach the final stage: solving for the remaining zeros using the quadratic formula. The formula is \(x = [-b \pm \sqrt{b^2 -4ac}] / (2a)\), which finds the solutions for any quadratic equation of the form \(ax^2+bx+c=0\).

In the simplified version of our original polynomial, we end up with a quadratic \(2x^2-x-1=0\) after factoring out \(x+2\). Applying the quadratic formula, we calculate two more zeros: \(0.5\) and \(-1\). These solutions, in combination with the zero identified through synthetic division, give us all the zeros of the polynomial function.