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Answer :
Final Answer:
The polynomial 2x^4 - 7x^3 - 13x^2 + 63x - 45 can be factorized as (x - 1)(x - 3)(2x - 5)(x + 3).
Explanation:
To factorize the given polynomial using the factor theorem, we first look for its roots. By applying synthetic division or trial and error, we find that x = 1, x = 3, x = 5/2, and x = -3 are the roots of the polynomial, as substituting these values makes the polynomial equal to zero. Therefore, using the factor theorem, we can write the polynomial as (x - 1)(x - 3)(x - 5/2)(x + 3).
Next, we simplify the expression by multiplying the linear factors together. This gives us (x - 1)(x - 3)(2x - 5)(x + 3). To ensure correctness, we can expand this expression and compare it with the original polynomial to verify that they are equivalent.
Expanding (x - 1)(x - 3)(2x - 5)(x + 3), we combine like terms and simplify to obtain 2x^4 - 7x^3 - 13x^2 + 63x - 45, which matches the original polynomial. Thus, the correct factorization of the given polynomial is (x - 1)(x - 3)(2x - 5)(x + 3). (Option A)
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