Answer
When adding polynomials, the like terms [i.e. terms with the same power/exponent of variables] are combined. Since the degree of the polynomial is determined by the highest power of the variable in the polynomial, the resulting polynomial will have a degree equal to the largest degree among all the polynomials involved as the exponents of the variables remain unaffected while adding.
Work Step by Step
Let's use an example to put this into context. Take these two polynomials into consideration:
$P(x)=x^3+x^2+x$
$Q(x)=2x^2+x$
The first polynomial $P(x)$ has a degree of 3. This is because the highest power in the polynomial is 3. The highest power in the 2nd polynomial $Q(x)$, is 2 therefore, the degree of the polynomial $Q(x)$ is 2.
Based on the statement from the question, we expect that if $P(x)+Q(x)$ is evaluated, the result would have a degree of 3. Let's test this.
$(x^3+x^2+x)+(2x^2+x)$
Group the like terms:
$x^3+x^2+2x^2+x+x$
Then simplify:
$x^3+3x^2+2x$
$P(x)+Q(x)=x^3+3x^2+2x$
The highest power in the result is 3, therefore the degree of the result is 3. This proves that when adding polynomials the degree of the sum of two polynomials of different degrees equals the larger of their degrees.
In summary, when two polynomials of different degrees are added, the higher-degree term cannot be canceled or changed, so the degree of the sum equals the degree of the polynomial with the larger degree.
