Consider the product (2x−x3)(−3x4−7x2). Which of the following is the correct expression when each term of the first polynomial is distributed onto the second polynomial?(1 point)

Question

Responses

2x(−3x4−7x2)−x3(−3x4−7x2)
2 x left parenthesis negative 3 x superscript 4 baseline minus 7 x squared right parenthesis minus x cubed left parenthesis negative 3 x superscript 4 baseline minus 7 x squared right parenthesis

2x(−3x4−7x2)+x3(−3x4−7x2)
2 x left parenthesis negative 3 x superscript 4 baseline minus 7 x squared right parenthesis plus x cubed left parenthesis negative 3 x superscript 4 baseline minus 7 x squared right parenthesis

2x(−x3−3x4−7x2)
2 x left parenthesis negative 3 x cubed minus 3 x superscript 4 baseline minus 7 x squared right parenthesis

2x(−3x4)−x3(−7x2)
2 x left parenthesis negative 3 x superscript 4 baseline right parenthesis minus x cubed left parenthesis negative 7 x squared right parenthesis

Answer 1

To distribute each term of the first polynomial \((2x - x^3)\) onto the second polynomial \((-3x^4 - 7x^2)\), we need to apply the distributive property.

This means we take each term from the first polynomial and multiply it by each term in the second polynomial. The expression can be broken down as follows:

\[ (2x - x^3)(-3x^4 - 7x^2) = 2x(-3x^4) + 2x(-7x^2) - x^3(-3x^4) - x^3(-7x^2) \]

From this, we can identify the correct response according to the options provided:

1. \(2x(-3x^4 - 7x^2) - x^3(-3x^4 - 7x^2)\) matches our distribution.
2. \(2x(-3x^4 - 7x^2) + x^3(-3x^4 - 7x^2)\) does not match, as the signs are incorrect.
3. \(2x(-3x^3 - 3x^4 - 7x^2)\) does not match either, as it incorrectly combines terms.
4. \(2x(-3x^4) - x^3(-7x^2)\) is incomplete as it misses terms from the other part of the distribution.

Thus, the correct expression is:

\[ 2x(-3x^4 - 7x^2) - x^3(-3x^4 - 7x^2) \]

So the answer is the first option.