Question

which of the following is equivalent to $(p^{3})(2p^{2}-4p)(3p^{2}-1)$?
○ $(p^{3})(6p^{4}-12p^{3}-2p^{2}+4p)$
○ $(p^{3})(6p^{4}+4p)$
○ $(2p^{6}-4p^{3})(3p^{2}-1)$
○ $(2p^{5}-4p^{4})(3p^{5}-p^{3})$

Explanation:

Step1: Multiply the two binomials

Use the distributive property (FOIL method) on $(2p^2 - 4p)(3p^2 - 1)$:

$$\begin{align*} (2p^2)(3p^2) + (2p^2)(-1) + (-4p)(3p^2) + (-4p)(-1) &= 6p^4 - 2p^2 - 12p^3 + 4p\\ &= 6p^4 - 12p^3 - 2p^2 + 4p \end{align*}$$

Step2: Attach the remaining factor

Keep the $p^3$ factor outside the result from Step1:
$(p^3)(6p^4 - 12p^3 - 2p^2 + 4p)$

Step3: Verify other options (elimination)

• Option 2: Incorrect, missing middle terms from binomial multiplication.
• Option 3: Incorrect, multiplying $p^3$ with $(2p^2 - 4p)$ gives $2p^5 - 4p^4$, not $2p^6 - 4p^3$.
• Option 4: Incorrect, incorrectly distributes $p^3$ to both binomials and multiplies the results.

Answer:

$\boldsymbol{(p^3)(6p^4 - 12p^3 - 2p^2 + 4p)}$