Finding the highest degree in a polynomial sum with a constant multiplier

Key concept: Polynomial degree comparison decides whether the leading terms cancel or survive in $f(x)+a\cdot g(x)$.

Step by step

Identify the leading terms

The two polynomials are

$f(x)=x^4-3x^2+2$

and

$g(x)=2x^5-8x^4+11x^2-5.$

The expression is $f(x)+a\cdot g(x)$, where $a$ is a constant. The degree of a polynomial sum is determined by the highest-power term that does not cancel.

Here, the highest degree term in $f(x)$ is $x^4$, while the highest degree term in $g(x)$ is $2x^5$. Since $g(x)$ contains an $x^5$ term and $f(x)$ does not, the only way the sum could have degree 5 is if the coefficient of $x^5$ is nonzero.

Step-by-step degree analysis

Write the combined expression as

$f(x)+a\cdot g(x)=\left(x^4-3x^2+2\right)+a\left(2x^5-8x^4+11x^2-5\right).$

Distribute $a$:

$=2a x^5+(1-8a)x^4+(-3+11a)x^2+(2-5a).$

Now the degree depends on the coefficient of the highest power that remains after simplification. The $x^5$ term is $2a x^5$.

• If $a\neq 0$, then $2a\neq 0$, so the polynomial has degree 5.
• If $a=0$, then the expression becomes just $f(x)$, whose degree is 4.

Because the question asks for the largest possible degree, we choose any nonzero value of $a$ and get degree 5.

Why this works

The degree of a sum is not always the larger of the two degrees if cancellation occurs, but cancellation can only happen when the same power appears in both expressions. Since $f(x)$ has no $x^5$ term at all, there is nothing to cancel the $2a x^5$ term from $g(x)$ unless $a=0$.

That means the highest possible degree is achieved whenever the $x^5$ term survives. The lower terms may change with different values of $a$, but they do not affect the maximum degree.

Final answer

The largest possible degree of $f(x)+a\cdot g(x)$ is $\boxed{5}$.

Common mistake

A typical mistake is to compare only the degrees of $f$ and $g$ and say the answer must be 5 without checking the coefficient $a$. That is incomplete, because if $a=0$, the degree drops to 4. Another error is to assume that the $x^4$ term matters most because it appears in both polynomials. It does matter for special values of $a$, but it cannot beat an existing $x^5$ term. The correct approach is to expand the expression, inspect the leading term, and then decide whether it can vanish.

Pitfall alert

The main place this problem goes wrong is the leading term $2a x^5$. Students sometimes focus on the shared $x^4$ terms and try to force cancellation there, but the $x^5$ term dominates unless $a=0$. Another common slip is to interpret “largest possible degree” as the degree for every value of $a$; that is not what the question asks. If $a=0$, the degree is only 4, but the problem asks for the maximum over all constants. Keep the distinction between possible and guaranteed degrees clear, especially when one polynomial has a unique highest power that the other polynomial lacks.

Try different conditions

If the expression were $f(x)+a\cdot g(x)$ with $f(x)=x^5-3x^2+2$ and the same $g(x)=2x^5-8x^4+11x^2-5$, then the answer would depend on whether the leading terms cancel. Expanding gives $(1+2a)x^5-8a x^4+\cdots$. The degree is 5 for most values of $a$, but if $1+2a=0$, then $a=-\tfrac12$ and the $x^5$ term disappears, dropping the degree to 4. If the question asked for the smallest possible degree instead, you would look for that special cancelling value. This variant shows why checking leading coefficients is more important than looking only at the highest exponent.

Further reading

polynomial degree, leading term cancellation, constant multiple of polynomial