Question 19

x^7,-3x^5,-6x^3

The result of multiplying the polynomials (x² - 3) and (x^5 + 2x^3) is x^7 - 3x^5 - 6x^3. This outcome is derived from applying the distributive property to each term in the first polynomial with each term in the second polynomial.

A) x^7,-3x^5,-6x^3

This option correctly represents the result of the multiplication. When multiplying, the leading term x² from the first polynomial combines with x^5 from the second polynomial to produce x^7. Additionally, the -3 from the first polynomial multiplies with 2x^3 to yield -6x^3, and -3 multiplies with x^5 to give -3x^5. All terms are accurately represented in this choice.

B) x^10,2x^5,-6x^3

This option is incorrect as it contains the term x^10, which is not a product of the given polynomials. The highest degree term should come from x² multiplied by x^5, which results in x^7, not x^10. Furthermore, the coefficient of x^5 is incorrectly stated as 2x^5, rather than -3x^5.

C) 5x^5,2x^6,-6x^3

Option C is also incorrect. It incorrectly states the coefficient of x^5 as 5x^5, which misrepresents the product of the polynomials. The multiplication does not yield a term x^6 either; therefore, this option fails to match the result of the multiplication.

D) x^7,2x^5,-6

This option is incorrect as well. Although it correctly identifies x^7, it inaccurately lists 2x^5 instead of -3x^5. Moreover, the constant term should not be listed as -6; the multiplication of -3 and 2x^3 produces -6x^3, which is not reflected in this choice.

Conclusion

Option A is definitively correct as it accurately represents the result of multiplying the two polynomials, capturing all necessary terms and their coefficients. In contrast, all other options fail due to inaccuracies in the derived terms, demonstrating a misunderstanding of the multiplication process within polynomials.