What is the product of these two polynomials?

Rationale

x^3 - 8x² + 21x - 30

The product of the two given polynomials results in the expression x^3 - 8x² + 21x - 30, which is derived from applying the distributive property (also known as the FOIL method for binomials) to multiply each term correctly.

A (x^3 - 8x² + 21x - 30) CORRECT

This expression is the correct result of multiplying the two polynomials. Each term is derived accurately from the multiplication process, ensuring that all like terms are combined properly to yield the final polynomial.

B (x^3 - 8x² - 21x - 30) YOUR ANSWER

This choice inaccurately includes a negative sign in front of the 21x term. The correct multiplication yields a positive 21x, which is crucial for matching the coefficients from the original polynomials.

C (x^3 - 8x² - 9x - 30) YOUR ANSWER

This option miscalculates the linear term, presenting a -9x instead of the correct +21x. The error arises from an incorrect combination of terms that results from the polynomial multiplication process.

D (x^3 + 8x² + 21x + 30) YOUR ANSWER

This choice incorrectly alters the signs of the coefficients for both the x² and the constant terms. The original polynomials do not produce positive coefficients for these terms; thus, this polynomial does not reflect the correct product.

E (x^3 + 8x^2 - 9x + 30) YOUR ANSWER

Like option D, this expression presents incorrect signs. Specifically, it misrepresents the x² term as positive and the linear term as -9x, diverging from the correct multiplication results.

Conclusion

In polynomial multiplication, maintaining sign accuracy and correctly combining like terms is essential for arriving at the correct product. The expression x^3 - 8x² + 21x - 30 faithfully represents the product of the two original polynomials, while all other options contain critical errors in sign and coefficient representation. Understanding these fundamentals is key to mastering polynomial operations in algebra.