A polynomial has non-negative integer powers of $x$. Its degree is the highest power with a nonzer

Question

A polynomial has non-negative integer powers of $x$. Its degree is the highest power with a nonzero coefficient.

$\mathrm{P}(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\quad(a_n\ne0)$

A nonzero constant has degree zero. The zero polynomial has no degree under this convention.

Expression	Polynomial?	Degree	Value
$\mathrm{P}(x)=4x^4-5x^3-2x+1$	Yes	4	$\mathrm{P}(-1)=12$
$\mathrm{Q}(x)=x^6-x^3+x+2$	Yes	6	$\mathrm{Q}(2)=60$
$\mathrm{f}(x)=3x^2+\sqrt x-3x+1$	No	—	$\mathrm{f}(4)=39$
$\mathrm{g}(x)=10$	Yes	0	$\mathrm{g}(-1)=10$
$\mathrm{k}(x)=3x^{-2}-x^{-4}+5x^2-2$	No	—	$\mathrm{k}(-2)=\frac{299}{16}$

Adding and subtracting

Collect like powers. When subtracting, change the sign of every term in the second polynomial.

$(4x^5+3x^3-12x+7)-(4x^3-3x^2)=4x^5-x^3+3x^2-12x+7$

Accepted Answer

$\mathrm P$: polynomial, degree $4$, $\mathrm P(-1)=12$ $\mathrm Q$: polynomial, degree $6$, $\mathrm Q(2)=60$ $\mathrm f$: non-polynomial, $\mathrm f(4)=39$ $\mathrm g$: polynomial, degree $0$, $\mathrm g(-1)=10$ $\mathrm k$: non-polynomial, $\mathrm k(-2)=\frac{299}{16}$

Polynomials and Their Degrees

Polynomials and Their Degrees

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Solution:

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Expression	Polynomial?	Degree	Value
\(\mathrm{P}(x)=4x^4-5x^3-2x+1\)	Yes	4	\(\mathrm{P}(-1)=12\)
\(\mathrm{Q}(x)=x^6-x^3+x+2\)	Yes	6	\(\mathrm{Q}(2)=60\)
\(\mathrm{f}(x)=3x^2+\sqrt x-3x+1\)	No	—	\(\mathrm{f}(4)=39\)
\(\mathrm{g}(x)=10\)	Yes	0	\(\mathrm{g}(-1)=10\)
\(\mathrm{k}(x)=3x^{-2}-x^{-4}+5x^2-2\)	No	—	\(\mathrm{k}(-2)=\frac{299}{16}\)