a Consider the sum of a polynomial and it's opposite. Does the resulting expression fit the definition of a polynomial?
B
b How is a sum related to a difference?
A
a Yes, see solution.
B
b Yes, see solution.
Practice makes perfect
a We want to know if the sum of two polynomials is always a polynomial. To see if this is always true or not, let's consider any polynomial.
4x+7
This is a polynomial with 1 term. Now, let's add another polynomial to it! Again, any polynomial will do! Just as an example, let's pick a polynomial with 1 term.
4x+7 + 3x^2 = 3x^2+4x+7
The result has 2 + 1= 3 terms, but it is still a polynomial! Let's look at some other examples and consider what happens when we have like terms.
4x+7 + ( 5x^4+9x^2+7)
= 5x^4+12x^2+7
The result has 2 + 3 ≠3 terms! Some of the monomials combined together resulting in the same number of terms as that of the longest polynomial that we added. We have shown that the number of monomials can increase or stay the same. Is it possible for the number of monomials to decrease? YES!
4x+7 + ( - 4x+7) = 14
This resulting polynomial has 1 term because 4x+(- 4x) = 0. If this occurs to each of the terms in the polynomial the result will be 0 which is still a polynomial! Let's summarize what we have found when adding polynomials given that the longest polynomial being added has n terms.
The resulting polynomial could have more than n terms.
The resulting polynomial could have less than n terms.
The resulting polynomial cannot have less than 1 term.
Even though there are three scenarios, we have found the solution to the original exercise. When adding polynomials, the result is always a polynomial.
b We have been asked if the result is a polynomial when we subtract one polynomial from another. Consider the following example.
3x+7 - ( 3x^2+8)
Here, we have one polynomial being subtracted from another. Recall that subtraction means to add the opposite.
3x+7+(-3x^2-8)
Now, we have a sum of polynomials. From Part A, we know that the sum of polynomials is always a polynomial. Since the subtraction of one polynomial from another can be written as a sum, the difference is always a polynomial.
