Chapter A.3, Problem 139AYU

To determine

To find: Explain why the degree of the product of two nonzero polynomials equals the sum of their degrees.

Answer to Problem 139AYU

The polynomial is saying how many times the variable is multiplied by itself; $x^2$ is, obviously, $x\cdot x$ if you multiply a polynomial by another polynomial, the largest terms in each will be multiplied together and form the new largest term. If you take something times itself a certain number of times and multiply it by the same thing times itself a different number of times, then it just ends up multiplied by itself that many more times; $(x^2)*(x^2)=(x*x)*(x*x)=x^4$

$x^n * x^m = x^{n+m}$

Highest exponent of product is product of terms of highest degrees, say $m$ and $n$.

xxxxx $m$-times

xxxxxx $n$-times

Product has xxxxx xxxxxx $m+n$ times.

Explanation of Solution

The polynomial is saying how many times the variable is multiplied by itself; $x^2$ is, obviously, $x\cdot x$ if you multiply a polynomial by another polynomial, the largest terms in each will be multiplied together and form the new largest term. If you take something times itself a certain number of times and multiply it by the same thing times itself a different number of times, then it just ends up multiplied by itself that many more times; $(x^2)*(x^2)=(x*x)*(x*x)=x^4$

$x^n * x^m = x^{n+m}$

Highest exponent of product is product of terms of highest degrees, say $m$ and $n$.

xxxxx $m$-times

xxxxxx $n$-times

Product has xxxxx xxxxxx $m+n$ times.