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?

Question

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)−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

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(−3x4−7x2)+x3(−3x4−7x2)

Answer 1

To distribute the first polynomial \( (2x - x^3) \) onto the second polynomial \( (-3x^4 - 7x^2) \), we need to take each term from the first polynomial and multiply it by every term from the second polynomial.

This can be done as follows:

1. Distributing \( 2x \) over \( (-3x^4 - 7x^2) \): \[ 2x \cdot (-3x^4) + 2x \cdot (-7x^2) = -6x^5 - 14x^3 \]

2. Distributing \( -x^3 \) over \( (-3x^4 - 7x^2) \): \[ -x^3 \cdot (-3x^4) + -x^3 \cdot (-7x^2) = 3x^7 + 7x^5 \]

Combining these gives: \[ -6x^5 - 14x^3 + 3x^7 + 7x^5 = 3x^7 + (-6x^5 + 7x^5) - 14x^3 = 3x^7 + x^5 - 14x^3 \]

Now, to determine which of the provided expressions matches this distribution:

The closest expression is: \[ 2x(-3x^4 - 7x^2) + x^3(-3x^4 - 7x^2) \]

This represents correctly distributing each term of the first polynomial onto the second.

So the correct choice is: \[ 2x(-3x^4 - 7x^2) + x^3(-3x^4 - 7x^2) \]