Question

suppose you add or subtract two quadratic trinomials that use the same variable. what are the possible classifications for the sum or difference? explain.
select all that apply.
a. constant monomial
b. quadratic trinomial
c. quadratic binomial
d. linear binomial
e. linear trinomial
f. linear monomial
g. quadratic monomial

Explanation:

To determine the possible classifications when adding or subtracting two quadratic trinomials (of the form \(ax^{2}+bx + c\) and \(dx^{2}+ex + f\), where \(a
eq0\), \(d
eq0\)):

Step 1: Analyze the general form of the sum/difference

The sum (or difference) of two quadratic trinomials is \((ax^{2}+bx + c)\pm(dx^{2}+ex + f)=(a\pm d)x^{2}+(b\pm e)x+(c\pm f)\).

Step 2: Consider cases for coefficients

• Case 1: \(a = -d\)

Then the \(x^{2}\)-term cancels: \((a\pm d)x^{2}=0\), so the result is \((b\pm e)x+(c\pm f)\), which is a linear binomial (if \(b\pm e
eq0\) and \(c\pm f
eq0\)), linear monomial (if \(c\pm f = 0\) and \(b\pm e
eq0\)), or constant monomial (if \(b\pm e = 0\) and \(c\pm f
eq0\)).

• **Case 2: \(a

eq -d\)**
Then the \(x^{2}\)-term remains (\((a\pm d)
eq0\)), so the result is a quadratic trinomial (if \(b\pm e
eq0\) and \(c\pm f
eq0\)), quadratic binomial (if \(b\pm e = 0\) or \(c\pm f = 0\) but not both), or quadratic monomial (if \(b\pm e = 0\) and \(c\pm f = 0\)).

Step 3: Evaluate each option

• A. Constant monomial: Possible (e.g., \((x^{2}+x + 1)-(x^{2}+x)=1\)).
• B. Quadratic trinomial: Possible (e.g., \((x^{2}+x + 1)+(x^{2}+x + 1)=2x^{2}+2x + 2\)).
• C. Quadratic binomial: Possible (e.g., \((x^{2}+x + 1)+(x^{2}+0x + 1)=2x^{2}+x + 2\) – wait, no, correct example: \((x^{2}+x + 1)+(x^{2}-x + 1)=2x^{2}+2\) (quadratic binomial, since the \(x\)-term cancels)).
• D. Linear binomial: Possible (e.g., \((x^{2}+x + 1)-(x^{2}-x + 1)=2x\)? No, wait: \((x^{2}+x + 1)-(x^{2}+0x + 1)=x\) (linear monomial) or \((x^{2}+x + 1)-(x^{2}+x - 1)=2\) (constant monomial). Wait, correct linear binomial example: \((x^{2}+x + 1)-(x^{2}-x + 0)=2x + 1\) (linear binomial).
• E. Linear trinomial: Impossible. A linear trinomial would require three non - zero terms of degree 1, 0, but the sum/difference of two quadratics can have at most three terms (degree 2, 1, 0) or fewer. A linear trinomial would imply degree 1 with three non - zero terms, but the original polynomials are quadratic, so the \(x^{2}\)-term either exists (making it quadratic) or cancels (leaving at most two non - zero terms for linear/constant).
• F. Linear monomial: Possible (e.g., \((x^{2}+x + 1)-(x^{2}+0x + 1)=x\)).
• G. Quadratic monomial: Possible (e.g., \((x^{2}+0x + 0)+(x^{2}+0x + 0)=2x^{2}\)).

Answer:

A. constant monomial, B. quadratic trinomial, C. quadratic binomial, D. linear binomial, F. linear monomial, G. quadratic monomial