Question

Which of the following shows that polynomials are a closed system under addition?(1 point) Responses (0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=1.846x2−12.84x+5.96(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=1.846x2−12.84x+5.96 left parenthesis 0.923 x squared minus 6.42 x plus 2.98 right parenthesis plus left parenthesis 2.98 minus 6.42 x plus 0.923 x squared right parenthesis equals 1.846 x squared minus 12.84 x plus 5.96 (0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=2x2−13x+6(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=2x2−13x+6 left parenthesis 0.923 x squared minus 6.42 x plus 2.98 right parenthesis plus left parenthesis 2.98 minus 6.42 x plus 0.923 x squared right parenthesis equals 2 x squared minus 13 x plus 6 (0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=10.994x2+5.96(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=10.994x2+5.96 left parenthesis 0.923 x squared minus 6.42 x plus 2.98 right parenthesis plus left parenthesis 2.98 minus 6.42 x plus 0.923 x squared right parenthesis equals 10.994 x squared plus 5.96 (0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=3.903x2−12.84x+3.903

Answer 1

To determine which of the options shows that polynomials are a closed system under addition, we need to identify the option in which the sum of two polynomials results in another polynomial.

The first response is: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 1.846x^2 - 12.84x + 5.96 \]

The second response is: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 2x^2 - 13x + 6 \]

The third response is: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 10.994x^2 + 5.96 \]

The fourth response is: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 3.903x^2 - 12.84x + 3.903 \]

To meet the closure property for addition of polynomials, when we add two polynomials, the result must also be a polynomial.

Let's analyze the first response: \[ 0.923x^2 + 0.923x^2 + (-6.42x - 6.42x) + (2.98 + 2.98) = 1.846x^2 - 12.84x + 5.96 \] This adds correctly.

For the second response: \[ 0.923x^2 + 0.923x^2 + (-6.42x - 6.42x) + (2.98 + 2.98) = 2.846x^2 - 12.84x + 5.96 \] This does not equal to \(2x^2 - 13x + 6\) as proposed.

For the third response: \[ 0.923x^2 + 0.923x^2 + (-6.42x - 6.42x) + (2.98 + 2.98) = 1.846x^2 - 12.84x + 5.96 \] This does not equal to \(10.994x^2 + 5.96\) either.

Finally, for the fourth response: \[ 0.923x^2 + 0.923x^2 + (-6.42x - 6.42x) + (2.98 + 2.98) = 1.846x^2 - 12.84x + 5.96 \] Again, does not equal to \(3.903x^2 - 12.84x + 3.903\).

Therefore, the first option is correct as it properly sums two polynomials and gives a polynomial as the result: \[ \text{(1) } (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 1.846x^2 - 12.84x + 5.96 \]

This shows that polynomials are closed under addition.