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Given:

\[ P(x) = 3x^6 + 3x^4 - 3x + 5 \]

\[ Q(x) = 3x^6 + 5 \]

Calculate the product \( P(x) \cdot Q(x) \).

Options:

1) \( 9x^{12} - 9x^{11} - 9x^7 + 24x^6 - 15x^4 - 15x + 15 \)

2) \( 15x^{12} - 9x^{10} - 9x^7 - 24x^6 - 15x^4 - 15x + 15 \)

3) \( 9x^{12} + 9x^{10} - 9x^7 + 30x^6 + 15x^4 - 15x + 25 \)

4) \( 9x^{12} + 15x^{11} - 9x^7 + 24x^6 - 15x^4 + 15x + 15 \)

Answer :

We are given the polynomials

[tex]$$
P(x)=3x^6+3x^4-3x+5 \quad \text{and} \quad Q(x)=3x^6+5.
$$[/tex]

To calculate the product [tex]$P(x) \cdot Q(x)$[/tex], we will distribute each term of [tex]$P(x)$[/tex] with each term of [tex]$Q(x)$[/tex].

1. Multiply [tex]$3x^6$[/tex] by each term in [tex]$Q(x)$[/tex]:
- [tex]$3x^6 \cdot 3x^6 = 9x^{12}$[/tex]
- [tex]$3x^6 \cdot 5 = 15x^6$[/tex]

2. Multiply [tex]$3x^4$[/tex] by each term in [tex]$Q(x)$[/tex]:
- [tex]$3x^4 \cdot 3x^6 = 9x^{10}$[/tex]
- [tex]$3x^4 \cdot 5 = 15x^4$[/tex]

3. Multiply [tex]$-3x$[/tex] by each term in [tex]$Q(x)$[/tex]:
- [tex]$-3x \cdot 3x^6 = -9x^7$[/tex]
- [tex]$-3x \cdot 5 = -15x$[/tex]

4. Multiply [tex]$5$[/tex] by each term in [tex]$Q(x)$[/tex]:
- [tex]$5 \cdot 3x^6 = 15x^6$[/tex]
- [tex]$5 \cdot 5 = 25$[/tex]

Now, combine all these results together:

[tex]$$
\begin{aligned}
P(x) \cdot Q(x) &= 9x^{12} + 15x^6 + 9x^{10} + 15x^4 - 9x^7 - 15x + 15x^6 + 25 \\
&= 9x^{12} + 9x^{10} - 9x^7 + (15x^6 + 15x^6) + 15x^4 - 15x + 25 \\
&= 9x^{12} + 9x^{10} - 9x^7 + 30x^6 + 15x^4 - 15x + 25.
\end{aligned}
$$[/tex]

Thus, the product [tex]$P(x) \cdot Q(x)$[/tex] is

[tex]$$
9x^{12}+9x^{10}-9x^7+30x^6+15x^4-15x+25,
$$[/tex]

which corresponds to option 3.

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Rewritten by : Barada