Write the expression -6(7 - 3x^3) + 2x(x^2 - 3) as a polynomial in standard form.

Step-by-step solution

1. 1

   Distribute the first product, minding the signs. $-6\cdot 7=-42$ and $-6\cdot\left(-3x^{3}\right)=+18x^{3}$, because a negative times a negative is positive. So $-6\left(7-3x^{3}\right)=-42+18x^{3}.$

2. 2

   Distribute the second product. Multiplying $2x$ into the bracket uses the law $x\cdot x^{2}=x^{3}$: $2x\left(x^{2}-3\right)=2x^{3}-6x .$

3. 3

   Write everything on one line. $-42+18x^{3}+2x^{3}-6x .$

4. 4

   Collect like terms. Only the cubic terms match: $18x^{3}+2x^{3}=20x^{3}$. The terms $-6x$ and $-42$ have no partners.

5. 5

   Order by descending degree and check. Standard form is $20x^{3}-6x-42 .$ Testing $x=1$: the original gives $-6(7-3)+2(1-3)=-24-4=-28$, and the answer gives $20-6-42=-28$, so the expansion is correct.