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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].
### 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].
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