Final answer:
The factorization of the polynomial 10x² + 3x - 27 to (2x - 3)(5x + 9) is not a valid factorization because when multiplied out, it does not give the original polynomial.
Step-by-step explanation:
A student has factored the polynomial 10x² + 3x - 27 to (2x - 3)(5x + 9). We are asked which statement about this factorization is true. We should evaluate each statement provided. First, we know that factorization is the process of breaking down a polynomial into simpler parts, or 'factors', that when multiplied together give back the original polynomial. This is different from expanding, which is the process of multiplying the factors to get the polynomial.
Secondly, a correct factorization should have the same roots, that is the values of x that make the polynomial equal zero, as the original polynomial. This follows from the fundamental theorem of algebra which states that polynomial equations have solutions, or roots, corresponding to the factors of the polynomial.
Therefore, the statement 'It has the same roots as the original polynomial' (option b) is true, provided that the factorization has been done correctly. This, however, can be verified by multiplying the factors back out. Upon doing so, we realize that the factorization is incorrect because the terms do not multiply to give the original polynomial. Hence, the correct statement is that it is 'not a valid factorization' of the polynomial (option c).