Answered

What is the additive inverse of the polynomial $-7 y^2+x^2 y-3 x y-7 x^2$?


A$7 y^2-x^2 y+3 x y+7 x^2$
B$-7 y^2-x^2 y-3 x y-7 x^2$
C$7 y^2+x^2 y-3 x y-7 x^2$
D$7 y^2+x^2 y+3 x y+7 x^2$

Verified Answer :

  • The additive inverse of a polynomial \(P\) is found by calculating \(-P\).
  • Multiply every term in \(-7y^2 + x^2y - 3xy - 7x^2\) by \(-1\).
  • This results in \(7y^2 - x^2y + 3xy + 7x^2\).
  • The final answer is \(\boxed{a}\).

Explanation

  1. Understanding the Additive Inverse

The problem asks for the additive inverse of the polynomial \(P = -7y^2 + x^2y - 3xy - 7x^2\). By definition, the additive inverse \(Q\) of a polynomial \(P\) is the expression that, when added to \(P\), results in zero. Mathematically, this means \(P + Q = 0\), which implies \(Q = -P\). To find the additive inverse, we negate each term of the original polynomial.

  1. Calculating the Additive Inverse

To find the additive inverse \(-P\), we distribute the negative sign across all terms of \(P\):
\(-P = -(-7y^2 + x^2y - 3xy - 7x^2)\)
This equates to:
\(-P = -(-7y^2) - (x^2y) - (-3xy) - (-7x^2)\)
Applying the rules of sign multiplication, where a negative multiplied by a negative is a positive, and a negative multiplied by a positive is a negative, we get:
\(-P = 7y^2 - x^2y + 3xy + 7x^2\)

  1. Identifying the Correct Option

Comparing the result \(7y^2 - x^2y + 3xy + 7x^2\) with the provided options:

A\(7y^2 - x^2y + 3xy + 7x^2\)
B\(7y^2 + x^2y - 3xy - 7x^2\)
C\(-7y^2 - x^2y - 3xy - 7x^2\)
D\(7y^2 + x^2y + 3xy + 7x^2\)
E\(7y^2 - x^2y + 3xy + 7x^2\)

The calculated expression matches option a (and e). Thus, the correct additive inverse is \(7y^2 - x^2y + 3xy + 7x^2\), which corresponds to option a.

Examples

The concept of additive inverses is foundational in algebra, similar to the concept of opposites in arithmetic. For instance, if you represent a debt of \$500 as @@MATH_TOKEN_24@@, the additive inverse, which is @@MATH_TOKEN_25@@, represents gaining \$500 to bring your balance to zero. In physics, when balancing equations for static equilibrium, you often find the necessary force that counteracts existing forces, essentially finding the additive inverse of the sum of the acting force vectors to ensure the net force is zero. Understanding these opposites allows for more efficient problem solving and structural analysis in both theoretical and practical applications.

Related Questions