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The Rational Root Theorem is a useful tool to identify possible rational roots of polynomial equations. It states that any potential rational root of a polynomial equation \[P(x) = a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_1 x + a_0\] must be a factor of the constant term \(a_0\) divided by a factor of the leading coefficient \(a_n\).To illustrate, consider our polynomial \(f(x)=x^4+2x^3-3x^2+24x-180\). Here, the constant term is \(-180\) and the leading coefficient is \(1\).

• The factors of \(-180\) are: ±1, ±2, ±3, ±4, ±5, ±6, ±9, ±10, ±12, ±15, ±18, ±20, ±30, ±36, ±45, ±60, ±90, ±180
• The factors of \(1\) are: ±1

Hence, the potential rational roots can be listed as the factors of \(-180\) over the factors of \(1\), which result in the possible values: ±1, ±2, ±3, ±4, ±5, ±6, ±9, ±10, ±12, ±15, ±18, ±20, ±30, ±36, ±45, ±60, ±90, ±180.Identifying these potential roots is essential to finding actual rational roots.

After listing the possible rational roots, synthetic division is an efficient method to test which values are roots of the polynomial.Unlike long division, synthetic division uses only the coefficients to perform the division.For example, to test if 6 is a root of our polynomial \(f(x)\), we set up synthetic division as follows:\[\begin{array}{r|rrrrr}6 & 1 & 2 & -3 & 24 & -180 \ & & 6 & 48 & 270 & 112 \hline & 1 & 8& 45 & 0 & \end{array}\]Here:

• We bring the first coefficient (1) down as-is.
• Multiply it by 6 and write it under the next coefficient (2).
• Add these values, continue multiplying the result by 6 and writing below the subsequent coefficients.

When the remainder is zero (as in this case), it confirms that 6 is a root.

Once the polynomial is reduced, if a quadratic equation remains, the quadratic formula can be used to find its roots.The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Let's apply it to our reduced quadratic polynomial \(x^2 + 5x + 10\):

• Here, \(a = 1\)
• \(b = 5\)
• \(c = 10\)

Substituting these values into the formula gives:\[x = \frac{-5 \pm \sqrt{25 - 40}}{2}\]Simplifying further:\[x = \frac{-5 \pm \sqrt{-15}}{2}\]Since \(-15\) under the square root results in an imaginary number, we simplify it to get:\[x = \frac{-5 \pm i\sqrt{15}}{2}\]Thus, the roots are complex: \(x = -\frac{5}{2} + \frac{\sqrt{15}}{2}i\) and \(x = -\frac{5}{2} - \frac{\sqrt{15}}{2}i\).

Factoring polynomials involves breaking down a polynomial into simpler polynomials that multiply together to give the original polynomial.After identifying a rational root using synthetic division or the Rational Root Theorem, the polynomial is 'depressed' (reduced) by that factor, simplifying it further.When factoring \(f(x)=x^4+2x^3-3x^2+24x-180\), after finding the root \(x = 6\) and depressing the polynomial, we rewrite it as:\[(x - 6)(x^3 + 8x^2 + 45x + 30)\]Next, we repeat the process to factor the cubic polynomial \(x^3 + 8x^2 + 45x + 30\). By testing further rational roots and using synthetic division, we find \(x = -3\):\[(x - 6)(x + 3)(x^2 + 5x + 10)\]Factoring simplifies solving higher-degree polynomials and finding their roots, revealing all rational and complex zeros in a structured manner.