Question 37

x³ - 8x² + 21x - 30

The product of the two polynomials (x - 5)(x² - 3x + 6) simplifies to x³ - 8x² + 21x - 30 through the distributive property.

A) x³ - 8x² + 21x - 30

This option is correct as it accurately represents the result of multiplying the two polynomials. By distributing (x - 5) across (x² - 3x + 6), we obtain each term: x*x², x*(-3x), x*6, -5*x², -5*(-3x), and -5*6, which combine to yield x³ - 8x² + 21x - 30.

B) x³ - 8x² - 21x - 30

This option is incorrect because it has the wrong sign for the linear term. The correct multiplication leads to a positive 21x, not a negative value, indicating a fundamental error in the distribution process.

C) x³ - 8x² - 9x - 30

This option is also incorrect. While the polynomial starts correctly with x³ - 8x², the linear term is inaccurately represented as -9x instead of the correct positive 21x, which reflects an incorrect computation during the distribution.

D) x³ + 8x² + 21x + 30

This option is incorrect because it has a positive sign for both the quadratic and constant terms, contradicting the proper distribution. The signs of the terms should reflect the polynomial's components resulting from the distribution of (x - 5).

E) x³ + 8x² - 9x + 30

This option is incorrect as well. It misrepresents the signs for the quadratic and linear terms, leading to a completely inaccurate polynomial. The correct multiplication does not yield positive coefficients for both the x² and the constant terms.

Conclusion

The correct answer, x³ - 8x² + 21x - 30, accurately reflects the result of multiplying the two given polynomials. All other options fail due to incorrect signs or miscalculations in the coefficients, demonstrating a misunderstanding of polynomial multiplication.