Faith can not be a work because it is a gift of grace:

Faith is enacted by God, by his will.

John 1:12-13:  But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name: 13Which were born, not of blood, nor of the will of the flesh, nor of the will of man, but of God

It is a gift given the children of disobedience.

Ephesians 2:1-5:  And you hath he quickened, who were dead in trespasses and sins; ²Wherein in time past ye walked according to the course of this world, according to the prince of the power of the air, the spirit that now worketh in the children of disobedience: ³Among whom also we all had our conversation in times past in the lusts of our flesh, fulfilling the desires of the flesh and of the mind; and were by nature the children of wrath, even as others. ⁴But God, who is rich in mercy, for his great love wherewith he loved us, ⁵Even when we were dead in sins, hath quickened us together with Christ, (by grace ye are saved;)

Something that is worked for is not received as gift; but a payment. If it must be worked for, then it is given out of obligation, out of debt.

Romans 4:4-5:  Now to him that worketh is the reward not reckoned of grace, but of debt. ⁵But to him that worketh not, but believeth on him that justifieth the ungodly, his faith is counted for righteousness.

Faith is a gift of grace, so it cannot be from works.

Romans 11:5-6:  Even so then at this present time also there is a remnant according to the election of grace. ⁶And if by grace, then is it no more of works: otherwise grace is no more grace. But if it be of works, then is it no more grace: otherwise work is no more work.