College

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

[tex]\left(2x^7 + 4x^6 + 3x^2 + 2x\right) + \left(3x^7 + 5x^6 + 2x^2 + 6x\right)[/tex]

A. [tex]x^7 + 9x^6 + 5x^2 + 8x[/tex]

B. [tex]5x^7 + 10x^6 + 5x^2 + 7x[/tex]

C. [tex]x^7 + 9x^6 + 5x^2 + 4x[/tex]

D. [tex]5x^7 + 9x^6 + 5x^2 + 8x[/tex]

Answer :

To find the sum of the two polynomials

[tex]$$
P(x) = 2x^7 + 4x^6 + 3x^2 + 2x \quad \text{and} \quad Q(x) = 3x^7 + 5x^6 + 2x^2 + 6x,
$$[/tex]

we add the coefficients of like terms.

1. For the [tex]$x^7$[/tex] terms:
[tex]$$
2x^7 + 3x^7 = (2+3)x^7 = 5x^7.
$$[/tex]

2. For the [tex]$x^6$[/tex] terms:
[tex]$$
4x^6 + 5x^6 = (4+5)x^6 = 9x^6.
$$[/tex]

3. For the [tex]$x^2$[/tex] terms:
[tex]$$
3x^2 + 2x^2 = (3+2)x^2 = 5x^2.
$$[/tex]

4. For the [tex]$x$[/tex] terms:
[tex]$$
2x + 6x = (2+6)x = 8x.
$$[/tex]

Putting these results together, the sum of the polynomials is:

[tex]$$
5x^7 + 9x^6 + 5x^2 + 8x.
$$[/tex]

This expression matches option D.

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