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

[tex]\left(6x^3 + 2x^2 + 9x\right) + \left(-2x^3 + 5x^2 + 3x\right)[/tex]

A. [tex]4x^3 + 7x^2 + 6x[/tex]
B. [tex]4x^3 + 7x^2 + 12x[/tex]
C. [tex]4x^3 + 7x^2 + 12[/tex]
D. [tex]8x^3 + 7x^2 + 12x[/tex]

Answer :

We are given the following two polynomials:

[tex]$$
6x^3 + 2x^2 + 9x \quad \text{and} \quad -2x^3 + 5x^2 + 3x.
$$[/tex]

Step 1. Add the coefficients for the [tex]$x^3$[/tex] terms.

[tex]$$
6x^3 + (-2x^3) = (6 - 2)x^3 = 4x^3.
$$[/tex]

Step 2. Add the coefficients for the [tex]$x^2$[/tex] terms.

[tex]$$
2x^2 + 5x^2 = (2 + 5)x^2 = 7x^2.
$$[/tex]

Step 3. Add the coefficients for the [tex]$x$[/tex] terms.

[tex]$$
9x + 3x = (9 + 3)x = 12x.
$$[/tex]

Step 4. Combine the results to write the final polynomial.

[tex]$$
4x^3 + 7x^2 + 12x.
$$[/tex]

The expression that represents the sum of the given polynomials is therefore:

[tex]$$
4x^3 + 7x^2 + 12x,
$$[/tex]

which corresponds to option B.

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