College

Which of the following represents the difference of the polynomials below?

[tex]\left(x^7 + 12x^6 + 7x^3\right) - \left(9x^7 + 8x^6 + 4x^3\right)[/tex]

A. [tex]-8x^7 + 4x^6 + 11x^3[/tex]
B. [tex]10x^7 + 20x^6 + 11x^3[/tex]
C. [tex]10x^7 + 20x^6 + 13x^3[/tex]
D. [tex]-8x^7 + 4x^6 + 3x^3[/tex]

Answer :

We start with the expression

[tex]$$
\left(x^7+12x^6+7x^3\right)-\left(9x^7+8x^6+4x^3\right).
$$[/tex]

First, distribute the negative sign to the second polynomial:

[tex]$$
x^7 + 12x^6 + 7x^3 - 9x^7 - 8x^6 - 4x^3.
$$[/tex]

Next, combine like terms:

1. For [tex]$x^7$[/tex]:
[tex]$$
x^7 - 9x^7 = -8x^7.
$$[/tex]

2. For [tex]$x^6$[/tex]:
[tex]$$
12x^6 - 8x^6 = 4x^6.
$$[/tex]

3. For [tex]$x^3$[/tex]:
[tex]$$
7x^3 - 4x^3 = 3x^3.
$$[/tex]

So, the resulting polynomial is

[tex]$$
-8x^7 + 4x^6 + 3x^3.
$$[/tex]

Thus, the correct answer is option D.

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