True or false, subtracting polynomials will yield a polynomial if and only if their leading coefficients are opposites. If false correct the underline words, if and only if their leading...

True or false, subtracting polynomials will yield a polynomial if and only if their leading coefficients are opposites. If false correct the underline words, if and only if their leading coefficients are opposites.

Asked on by knihi1978

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txmedteach's profile pic

txmedteach | High School Teacher | (Level 3) Associate Educator

Posted on

The question here seems a bit vague, but as I understand it, you are asking whether the following statement is true: 

Subtracting polynomials will yield a polynomial if and only if their leading coefficients are additive inverses.

The best way to test "if and only if" statements is to see if they are correct in the "if" case (coming up with a general example where the condition stated is met). Then, test it in the opposite direction to see if the statement only holds if the condition is met.

Try the "if" way first:

`A(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0`

`B(x) = -a_nx^n-a_(n-1)x^(n-1) - ... -a_1x-a_0`

`A(x) - B(x) = 2a_nx^n + 2a_(n-1)x^(n-1) + ... + 2a_1x + 2a_0`

Clearly, the subtraction yields a true statement, that when the coefficients are additive inverses the result of subtraction yields a polynomial.

Now, we'll see if the statement holds when the condition is not met:

` `

`A(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0`

`B(x) = b_nx^n+b_(n-1)x^(n-1) + ... + b_1x + b_0`

`A(x) - B(x) = (a_n-b_n)x^n + (a_(n-1)-b_(n-1))x^(n-1) + ... + (a_1-b_1)x + (a_0-b_0)`

Clearly, the subtraction here will yield a polynomial, too! Therefore, the "only if" portion of the statement is false.

To get the correct form of the statement, we remove the "only if" language to get the following correct statement:

Subtracting polynomials will yield a polynomial if their leading coefficients are additive inverses.

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nick-teal's profile pic

nick-teal | High School Teacher | (Level 3) Adjunct Educator

Posted on

To add to this answer:

You can just say "subtracting polynomials will result in a polynomial."

It is true that if the leading terms are opposite subtracting two polynomails will lead to another polynomial, but the first two terms are not required to be opposite.

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