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The Rational Zero Theorem is a handy tool in polynomial factorization. It states that if you have a polynomial with integer coefficients, any possible rational zero will be a fraction of the factors of the constant term divided by the factors of the leading coefficient. For example, for the polynomial \(x^3 - x^2 - 21x + 45\), the factors of the constant term (45) could be \( \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45\). Since the leading coefficient is 1 in this case (factors: \( \pm 1 \)), the possible rational roots of \(x^3 - x^2 - 21x + 45\) are just these factors. This step helps narrow down the potential solutions that you need to test, making the problem much more manageable.

Synthetic division is a simplified form of polynomial division, useful for testing potential roots. It's faster and less error-prone than long division. After listing the possible rational roots using the Rational Zero Theorem, synthetic division is used to see which of these make the polynomial equal to zero. For instance, testing \(x=3\) for the polynomial \(x^3 - x^2 - 21x + 45\) involves setting up a synthetic division:

• Write down the coefficients: \(1, -1, -21, 45\)
• Place 3 on the left side.
• Bring down the leading coefficient (1).
• Multiply the number you brought down by 3 and put the result under the next coefficient: \(1 * 3 = 3\)
• Add this to the next coefficient: \(-1 + 3 = 2\).
• Repeat the process until you get the final sum (which should be 0 if 3 is indeed a root).

This division process makes it evident if \(x=3\) is a valid root. For our polynomial, it turns out 3 is a valid root because the remainder is zero.

Once you've identified a root using synthetic division, the next step is to factor out the found root from the polynomial. For \(x^3 - x^2 - 21x + 45\) and the root \(x = 3\), synthetic division gives us the quotient \(x^2 + 2x - 15\). This allows us to rewrite the original polynomial as:

• \((x - 3)(x^2 + 2x - 15)\)

Testing and confirming all valuable findings ensure detailed and thorough polynomial factorization. We have now broken the original polynomial into a product of binomials. The new quadratic term can also be factored further to get the complete factorization. Factoring \(x^2 + 2x - 15\) yields \((x + 5)(x - 3)\), resulting in \((x - 3)(x - 3)(x + 5)\) or \((x-3)^2(x+5)\).

With the polynomial already broken into simpler parts, we may still need to factor quadratics further. As seen, factorizing \(x^2 + 2x - 15\) involves finding two numbers that multiply to -15 and add to 2. These numbers are 5 and -3. Thus, we split the middle term and factor by grouping:

• Rewrite the quadratic: \(x^2 + 2x - 15 = x^2 + 5x - 3x - 15\).
• Group the terms: \((x^2 + 5x) + (-3x - 15)\).
• Factor out the common terms: \(x(x + 5) - 3(x + 5)\).
• Combine the common factors: \((x - 3)(x + 5)\).

By factorizing the quadratic term, the complete polynomial factorization of \(x^3 - x^2 - 21x + 45\) is \((x-3)^2(x+5)\). This makes it easier to carry out operations like partial fraction decomposition, foundational for various calculus applications.