91Ó°ÊÓ

The Rational Root Theorem is an essential tool in algebra that helps identify potential rational zeros of a polynomial. The theorem states that any rational zero of a polynomial of the form \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \] will be a fraction \( \frac{p}{q} \), where:

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• \( p \) is a factor of the constant term \(a_0\).
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• \( q \) is a factor of the leading coefficient \(a_n\).

For the polynomial in our example, \(f(x) = 15x^3 + 61x^2 + 2x - 8\), we list the factors of the constant term (-8) and the leading coefficient (15), as shown:

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• Factors of -8: ±1, ±2, ±4, ±8
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• Factors of 15: ±1, ±3, ±5, ±15

This results in possible rational zeros: ±1, ±2, ±4, ±8, ±1/3, ±2/3, ±4/3, ±8/3, ±1/5, ±2/5, ±4/5, ±8/5, ±1/15, ±2/15, ±4/15, ±8/15. Listing all these possible zeros is the first crucial step in finding actual rational zeros.

Identifying actual rational zeros from the list of possible rational zeros requires us to evaluate the polynomial at each potential zero. This process involves substituting each candidate into the polynomial function and determining whether the result equals zero. This method can be slightly tedious but is systematic:For example:

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• \(f(1) = 15(1)^3 + 61(1)^2 + 2(1) - 8 = 15 + 61 + 2 - 8 = 70 eq 0\)
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• \(f(-1) = 15(-1)^3 + 61(-1)^2 + 2(-1) - 8 = -15 + 61 - 2 - 8 = 36 eq 0\)
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• \(f(2) = 15(2)^3 + 61(2)^2 + 2(2) - 8 = 120 + 244 + 4 - 8 = 360 eq 0\)

Continuing this process, we find that \( f(1/3) = 0 \) and \( f(-8/15) = 0 \). Thus, the rational zeros of \(15x^3 + 61x^2 + 2x - 8\) are \(1/3\) and \(-8/15\). These zeros mean we can now factor the polynomial accordingly.

After identifying the rational zeros, we move on to factoring the polynomial into linear factors using polynomial division. Given our zeros \(1/3\) and \(-8/15\), we can form the linear factors \(3x - 1\) and \(15x + 8\). The polynomial \(f(x) = 15x^3 + 61x^2 + 2x - 8\) can initially be expressed as:\[ f(x) = (3x - 1)(15x^2 + 66x + 24) \]We confirm this by performing polynomial division to revise the remaining polynomial factor \(15x^2 + 66x + 24\). This polynomial can further be factored as \( (5x + 4)(x + 2) \), so:\[ f(x) = (3x - 1)(5x + 4)(x + 2) \]Polynomial division helps verify the accuracy of our factorization and ensures there are no mistakes. By understanding these methods, you can solve similar polynomial problems efficiently.