standard form of a polynomial

Standard form is just a tidy way of writing a polynomial: line the terms up from the highest power down to the lowest, like books shelved tallest-to-shortest. So 7 - 5x + 3x^2 becomes 3x^2 - 5x + 7 in standard form.

Writing in standard form makes the polynomial easy to read and easy to work with. The degree is now the exponent on the very first term, the leading coefficient is the first number you see, and the constant sits politely at the end. Long division and synthetic division both assume the dividend is in this order.

Standard form does not change the polynomial's value — it is the same expression, just rearranged using the commutative property of addition. Before you do almost any operation (adding, dividing, factoring), putting things in standard form first prevents careless slips.