Answer :
To calculate the product [tex]\( P(x) \cdot Q(x) \)[/tex], we first need to expand the multiplication of the two polynomials. Here are the given polynomials:
[tex]\[ P(x) = 4x^7 + 3x^4 - 4x + 4y \][/tex]
[tex]\[ Q(x) = 3x^7 + 3 \][/tex]
Now, let's proceed with the multiplication term by term to find all the resulting terms:
1. Multiply each term in [tex]\( P(x) \)[/tex] by [tex]\( 3x^7 \)[/tex]:
[tex]\[ (4x^7) \cdot (3x^7) = 12x^{14} \][/tex]
[tex]\[ (3x^4) \cdot (3x^7) = 9x^{11} \][/tex]
[tex]\[ (-4x) \cdot (3x^7) = -12x^8 \][/tex]
[tex]\[ (4y) \cdot (3x^7) = 12x^7 y \][/tex]
2. Multiply each term in [tex]\( P(x) \)[/tex] by [tex]\( 3 \)[/tex]:
[tex]\[ (4x^7) \cdot 3 = 12x^7 \][/tex]
[tex]\[ (3x^4) \cdot 3 = 9x^4 \][/tex]
[tex]\[ (-4x) \cdot 3 = -12x \][/tex]
[tex]\[ (4y) \cdot 3 = 12y \][/tex]
Now, combine all these terms:
[tex]\[ 12x^{14} + 9x^{11} - 12x^8 + 12x^7 y + 12x^7 + 9x^4 - 12x + 12y \][/tex]
This is the expanded form of the product [tex]\( P(x) \cdot Q(x) \)[/tex].
So the correct option is not listed among the given choices. Let's re-evaluate each step closely to confirm this is correct:
[tex]\[ 4x^7 \cdot 3x^7 = 12x^{14} \][/tex]
[tex]\[ 3x^4 \cdot 3x^7 = 9x^{11} \][/tex]
[tex]\[ -4x \cdot 3x^7 = -12x^8 \][/tex]
[tex]\[ 4y \cdot 3x^7 = 12x^7 y \][/tex]
and
[tex]\[ 4x^7 \cdot 3 = 12x^7 \][/tex]
[tex]\[ 3x^4 \cdot 3 = 9x^4 \][/tex]
[tex]\[ -4x \cdot 3 = -12x \][/tex]
[tex]\[ 4y \cdot 3 = 12y \][/tex]
Reconfirming each product step yields the same polynomial:
[tex]\[ 12x^{14} + 9x^{11} - 12x^8 + 12x^7 y + 12x^7 + 9x^4 - 12x + 12y \][/tex]
Therefore, the result of the multiplication [tex]\( P(x) \cdot Q(x) \)[/tex] is:
[tex]\[ \boxed{12x^{14} + 9x^{11} - 12x^8 + 12x^7 y + 12x^7 + 9x^4 - 12x + 12y} \][/tex]
[tex]\[ P(x) = 4x^7 + 3x^4 - 4x + 4y \][/tex]
[tex]\[ Q(x) = 3x^7 + 3 \][/tex]
Now, let's proceed with the multiplication term by term to find all the resulting terms:
1. Multiply each term in [tex]\( P(x) \)[/tex] by [tex]\( 3x^7 \)[/tex]:
[tex]\[ (4x^7) \cdot (3x^7) = 12x^{14} \][/tex]
[tex]\[ (3x^4) \cdot (3x^7) = 9x^{11} \][/tex]
[tex]\[ (-4x) \cdot (3x^7) = -12x^8 \][/tex]
[tex]\[ (4y) \cdot (3x^7) = 12x^7 y \][/tex]
2. Multiply each term in [tex]\( P(x) \)[/tex] by [tex]\( 3 \)[/tex]:
[tex]\[ (4x^7) \cdot 3 = 12x^7 \][/tex]
[tex]\[ (3x^4) \cdot 3 = 9x^4 \][/tex]
[tex]\[ (-4x) \cdot 3 = -12x \][/tex]
[tex]\[ (4y) \cdot 3 = 12y \][/tex]
Now, combine all these terms:
[tex]\[ 12x^{14} + 9x^{11} - 12x^8 + 12x^7 y + 12x^7 + 9x^4 - 12x + 12y \][/tex]
This is the expanded form of the product [tex]\( P(x) \cdot Q(x) \)[/tex].
So the correct option is not listed among the given choices. Let's re-evaluate each step closely to confirm this is correct:
[tex]\[ 4x^7 \cdot 3x^7 = 12x^{14} \][/tex]
[tex]\[ 3x^4 \cdot 3x^7 = 9x^{11} \][/tex]
[tex]\[ -4x \cdot 3x^7 = -12x^8 \][/tex]
[tex]\[ 4y \cdot 3x^7 = 12x^7 y \][/tex]
and
[tex]\[ 4x^7 \cdot 3 = 12x^7 \][/tex]
[tex]\[ 3x^4 \cdot 3 = 9x^4 \][/tex]
[tex]\[ -4x \cdot 3 = -12x \][/tex]
[tex]\[ 4y \cdot 3 = 12y \][/tex]
Reconfirming each product step yields the same polynomial:
[tex]\[ 12x^{14} + 9x^{11} - 12x^8 + 12x^7 y + 12x^7 + 9x^4 - 12x + 12y \][/tex]
Therefore, the result of the multiplication [tex]\( P(x) \cdot Q(x) \)[/tex] is:
[tex]\[ \boxed{12x^{14} + 9x^{11} - 12x^8 + 12x^7 y + 12x^7 + 9x^4 - 12x + 12y} \][/tex]
