Completely simplify the difference of the polynomials a^3b + 9a^2b^2 - 4ab^5 and a^3b - 3a^2b^2 + ab^5, then state how many terms the result has and what its degree is.

Step-by-step solution

1. 1

   Write the subtraction with brackets before doing anything else. “The difference of $P$ and $Q$” means $P-Q$ in that order, and the second polynomial must be bracketed:

   $\big(a^3b+9a^2b^2-4ab^5\big)-\big(a^3b-3a^2b^2+ab^5\big)$

   Forgetting the bracket and subtracting only the first term of $Q$ is the single most common error in this problem type, and it changes both the number of terms and the degree of the answer.

2. 2

   Distribute the minus sign across every term of the second polynomial. Each sign flips:

   $a^3b+9a^2b^2-4ab^5\;-\;a^3b\;+\;3a^2b^2\;-\;ab^5$

   Note that $-(-3a^2b^2)=+3a^2b^2$ and $-(+ab^5)=-ab^5$: two of the three signs change, which is why the middle terms will add rather than cancel.

3. 3

   Combine like terms. Terms are “like” only when both variables carry identical exponents:

   $a^3b-a^3b=0$

   $9a^2b^2+3a^2b^2=12a^2b^2$

   $-4ab^5-ab^5=-5ab^5$

   So the completely simplified difference is

   $12a^2b^2-5ab^5$

   The $a^3b$ terms were identical in both polynomials, so they annihilate — that is what drops the answer from three terms to two.

4. 4

   Count the terms: it is a binomial. After simplification exactly two unlike terms remain, $12a^2b^2$ and $-5ab^5$. They are not like terms ($a^2b^2$ versus $ab^5$), so no further combining is possible and the result is a binomial.

5. 5

   Find the degree by adding the exponents within each term, then taking the largest. For a term in several variables the degree is the sum of the exponents, not the largest single exponent:

   $\deg\!\left(12a^2b^2\right)=2+2=4,\qquad \deg\!\left(-5ab^5\right)=1+5=6$

   The degree of the polynomial is the maximum of these, so the degree is $6$. A frequent slip is reading the degree off the first term, which would give the wrong answer $4$; the terms are not written in descending degree order here.

6. 6

   State the conclusion. The completely simplified difference is $12a^2b^2-5ab^5$: a binomial of degree 6.