What is the product of the two polynomials: (x - 5)(x² - 3x + 6)?

Rationale

x³ - 8x² + 21x - 30

To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (also known as the FOIL method for binomials), which results in the correct polynomial expansion of x³ - 8x² + 21x - 30.

A (x³ - 8x² + 21x - 30) CORRECT

This choice correctly represents the product obtained by multiplying the polynomials. The distribution yields the following: x * (x² - 3x + 6) gives x³ - 3x² + 6x, and -5 * (x² - 3x + 6) results in -5x² + 15x - 30. Combining these results leads to the final expression x³ - 8x² + 21x - 30.

B (x³ - 8x² - 21x - 30) YOUR ANSWER

This option mistakenly includes a negative sign in front of the 21x term. The correct multiplication results in a positive 21x, not a negative, thus making this choice incorrect.

C (x³ - 8x² - 9x - 30) YOUR ANSWER

This choice has an incorrect coefficient for the x term. The correct calculation shows that the x term should sum to +21x, not -9x, leading to an incorrect polynomial.

D (x³ + 8x² + 21x + 30) YOUR ANSWER

This option contains all incorrect signs and terms. The leading coefficient of x² is incorrectly positive, and the constant term is also incorrectly positive. The signs do not match the result of the multiplication.

E (x³ + 8x² - 9x + 30) YOUR ANSWER

This choice also has incorrect signs and coefficients. Both the x² and constant terms are incorrectly positive, and the x term is incorrectly stated as -9x instead of +21x, making this choice false.

Conclusion

The correct product of the polynomials (x - 5)(x² - 3x + 6) is accurately expressed as x³ - 8x² + 21x - 30. All other options either miscalculate or misrepresent the terms resulting from the polynomial multiplication, emphasizing the importance of careful distribution in algebraic expressions.