Explain why all solutions must be non-zero for this polynomial function. An algebraic argument must be provided for full credit. Graphing by hand or with technology will be granted partial credit unless the evidence on the graph is supported with algebra.
\[
f(x)=10 x^{5}-15 x^{4}+12 x^{3}-18 x^{2}+2 x-3
\]

Step 1 ：The question asks to explain why all solutions must be non-zero for the given polynomial function. To answer this, we need to understand the fundamental theorem of algebra which states that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots.

Step 2 ：In this case, the polynomial is of degree 5, so it must have 5 roots. However, the question asks why all solutions must be non-zero. This is because the constant term of the polynomial (in this case, -3) is the product of the roots. If one of the roots was zero, the constant term would also be zero. But in this case, the constant term is -3, not zero. Therefore, all roots of the polynomial must be non-zero.

Step 3 ：The roots of the polynomial are complex numbers. None of them are zero, which confirms our algebraic argument that all solutions must be non-zero. The roots are approximately 1.5, 0.000000000000001+1j, 0.000000000000001-1j, -0.00000000000000035+0.4472136j, and -0.00000000000000035-0.4472136j. The small numbers close to zero are due to the numerical precision of the calculation and should be considered as non-zero.

Step 4 ：Final Answer: All solutions must be non-zero for the given polynomial function because the constant term of the polynomial is the product of the roots. If one of the roots was zero, the constant term would also be zero. But in this case, the constant term is -3, not zero. Therefore, all roots of the polynomial must be non-zero. This is confirmed by the roots calculated, which are all non-zero. Therefore, the final answer is \(\boxed{\text{All solutions must be non-zero}}\).