State whether the following statements are true of false. If they are true, give a short...

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Basic Math

State whether the following statements are true of false. Ifthey are true, give a short justification. If they are false, givea counterexample. For each of the following, P(x) is apolynomial.
(a) If P(x) has only even powers, and P(a) = 0 then x^2 ? a^2divides P(x).
(b) If P(x) has only odd powers, and P(a) = 0 then x^2 ? a^2divides P(x).
(c) If P(x) has only even powers, then P(x) has at least onereal root.
(d) If P(x) = a7x^7 + a6x^6 + a3x^3 + a0, where ai ? R, ai isnot equal to zero, then P(x) has at least one real root.
(e) If P(x) has only even powers, then P(x) has at least onecomplex root.
(f) If P(x) = a7x^7 + a6x^6 + a3x^3 + a0, where ai ? R, ai isnot equal to zero , then P(x) has at least one complex root.

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4.3 Ratings (576 Votes)

a The remainder theorem states that the remainder of the division of a polynomial displaystyle fxPx by a linear polynomial displaystyle xrxa is equal to See Answer

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