To determine which of the options shows that polynomials are a closed system under addition, we need to check the accuracy of each equation stated in the options. The statement of closure means that when we add two polynomials, the result should also be a polynomial.
Let's test one of the given expressions step by step:
-
Expression to add:
\[
(0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2)
\]
-
Combine like terms:
- For \(x^2\) terms:
\[
0.923x^2 + 0.923x^2 = 1.846x^2
\]
- For \(x\) terms:
\[
-6.42x - 6.42x = -12.84x
\]
- For constant terms:
\[
2.98 + 2.98 = 5.96
\]
-
Putting it all together:
\[
1.846x^2 - 12.84x + 5.96
\]
Now, let's compare the actual result with the options given:
-
First option:
\[
(0.923 x^2 - 6.42 x + 2.98) + (2.98 - 6.42 x + 0.923 x^2) = 10.994 x^2 + 5.96 \quad \text{(Incorrect)}
\]
-
Second option:
\[
(0.923 x^2 - 6.42 x + 2.98) + (2.98 - 6.42 x + 0.923 x^2) = 3.903 x^2 - 12.84 x + 3.903 \quad \text{(Incorrect)}
\]
-
Third option:
\[
(0.923 x^2 - 6.42 x + 2.98) + (2.98 - 6.42 x + 0.923 x^2) = 1.846 x^2 - 12.84 x + 5.96 \quad \text{(Correct)}
\]
-
Fourth option:
\[
(0.923 x^2 - 6.42 x + 2.98) + (2.98 - 6.42 x + 0.923 x^2) = 2 x^2 - 13 x + 6 \quad \text{(Incorrect)}
\]
Conclusion:
The third option correctly shows that the addition of two polynomials results in another polynomial, confirming that polynomials are a closed system under addition.
So, the correct response is:
(0.923x²−6.42x+2.98)+(2.98−6.42x+0.923x²)=1.846x²−12.84x+5.96.
1 answer