Constant Term of a Polynomial

The constant term is the term in a polynomial that has degree zero.

For example, consider the following polynomial written in standard form. It has four terms:

$$ x^3y^2 - 2xy + 3x + 5 $$

The constant term is +5, as it's the only term that has degree zero.

Why is the constant term considered a degree-zero term?

It's called degree zero because it can be interpreted as the coefficient of any variable or number raised to the zero power.

Recall that any nonzero number or variable raised to the power of zero equals 1.

For example: 3⁰ = 1, 12⁰ = 1, x⁰ = 1, and so on.

So, we can rewrite the constant term as the coefficient of x⁰, like this:

$$ x^3y^2 - 2xy + 3x + 5x^0 $$

This is an equivalent expression. The polynomial hasn’t changed; we’ve just rewritten it in a different form to highlight that 5 is indeed a degree-zero term.

This confirms that the constant term (5) has degree zero within the context of the polynomial.

Note: If a polynomial doesn’t include a constant term - for example, $$ x^2 + xy + y $$ - we can still express it in an equivalent form by adding a constant term equal to zero: $$ x^2 + xy + y + 0 $$.

And so on.