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Let [tex]P(x) = 10x^5 - 35x^4 + 22x^3 + 13x^2 + 4x + 4[/tex].

(a) Use the Rational Roots Theorem to find all possible rational roots of [tex]P(x)[/tex].

(b) Find all roots of [tex]P(x)[/tex].

Answer :

Final answer:

The possible rational roots of the polynomial can be determined using the Rational Roots Theorem, while finding all roots, including irrational or complex ones, may involve numerical methods such as the Newton-Raphson method after checking for rational roots.

Explanation:

To find all possible rational roots of the polynomial P(x) = 10x5 −35x4 +22x3 +13x2 +4x+4, we use the Rational Roots Theorem. This theorem states that any rational root, expressed in its lowest terms p/q, will have p as a factor of the constant term and q as a factor of the leading coefficient.

The factors of the constant term (4) are ±1, ±2, ±4. The factors of the leading coefficient (10) are ±1, ±2, ±5, and ±10. Therefore, the possible rational roots can be ±1, ±2, ±5, ±10, ±1/2, ±1/5, ±1/10, ±2/5, or any of these with a negative sign.

Finding all the roots of P(x), including irrational and complex roots, might require numerical methods such as the Newton-Raphson method or other root-finding algorithms, but we'd first check the rational roots found previously. Once confirmed, we can use synthetic division or polynomial division to factor P(x) and find the remaining roots.

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