Theorem

Degree of a product of polynomials

Algebra • Polynomial

Formula

de g (f g) = de g f + de g g

For non-zero polynomials

f

and

g

with real coefficients,

de g (f g) = de g f + de g g

, because the two leading terms multiply to a term with a non-zero coefficient; by contrast

de g (f + g) \leq max (de g f, de g g)

, with equality unless the leading terms cancel.

How to use it

1Add the degrees for a product; never multiply them.
2For a power, $de g (f^{k}) = k de g f$ .
3For a sum or a difference take the larger degree, but check first whether the leading terms cancel.

Worked example

de g f = 5

and

de g g = 2

, what are the degrees of

f g

and of

f^{2}

Step-by-step

1
$de g (f g) = 5 + 2 = 7$
Why: The leading terms multiply and cannot cancel.
2
$de g (f^{2}) = 2 \times 5 = 10$
Why: A power is a repeated product, so the degrees add $k$ times.

Answer

7

and

10

Watch out

Both polynomials must be non-zero — the zero polynomial has no degree. The rule needs the product of the leading coefficients to be non-zero, which over the reals it always is.

Also known as

degree of a product, add the degrees, degree of a polynomial product, degree of a power of a polynomial

Degree of a product of polynomials

Formula

Worked example

Appears in past papers

2019 (II)

2018 (I)

2017 (II)