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------------------------------------------------ Two polynomials are shown.

A: [tex]3x^3 + 7[/tex]
B: [tex]7x + 3[/tex]

What is the resulting polynomial of [tex]A \times B[/tex]?

A. [tex]21x^4 + 49x[/tex]
B. [tex]3x^3 + 7x + 10[/tex]
C. [tex]21x^4 + 9x^3 + 49x + 21[/tex]
D. [tex]21x^4 + 21x^3 + 21x + 21[/tex]

Answer :

To find the resulting polynomial of [tex]\( A \times B \)[/tex], where [tex]\( A = 3x^3 + 7 \)[/tex] and [tex]\( B = 7x + 3 \)[/tex], we need to perform polynomial multiplication. Let's do this step-by-step:

### Step 1: Distribute each term in [tex]\( A \)[/tex] to each term in [tex]\( B \)[/tex].

Multiply [tex]\( 3x^3 \)[/tex] by each term in [tex]\( B \)[/tex]:
- [tex]\( 3x^3 \times 7x = 21x^4 \)[/tex]
- [tex]\( 3x^3 \times 3 = 9x^3 \)[/tex]

Multiply [tex]\( 7 \)[/tex] by each term in [tex]\( B \)[/tex]:
- [tex]\( 7 \times 7x = 49x \)[/tex]
- [tex]\( 7 \times 3 = 21 \)[/tex]

### Step 2: Combine all these results together.

The resulting polynomial is obtained by adding up all these terms:

[tex]\[
21x^4 + 9x^3 + 49x + 21
\][/tex]

So, the resulting polynomial of [tex]\( A \times B \)[/tex] is:

- [tex]\( 21x^4 + 9x^3 + 49x + 21 \)[/tex]

This matches the third option: [tex]\( 21x^4 + 9x^3 + 49x + 21 \)[/tex].