Sure, let's solve the given polynomial multiplication step-by-step.
Given the polynomial equation:
[tex]\[
\left(7 - 8x^3 - \frac{3}{2}x + \frac{5}{2}x \right)\left(4x^3 - 3x + 5\right)
\][/tex]
First, we simplify the terms inside each of the polynomial expressions.
For polynomial \(A(x)\):
[tex]\[
A(x) = 7 - 8x^3 - \frac{3}{2}x + \frac{5}{2}x
\][/tex]
Combining like terms:
[tex]\[
A(x) = 7 - 8x^3 + x
\][/tex]
Next, for polynomial \(B(x)\):
[tex]\[
B(x) = 4x^3 - 3x + 5
\][/tex]
Now, we need to multiply these polynomials together. The multiplication of two polynomials is performed by distributing every term in \(A(x)\) to every term in \(B(x)\).
[tex]\[
(7 - 8x^3 + x)(4x^3 - 3x + 5)
\][/tex]
Distribute each term of \(7\):
[tex]\[
7 \cdot 4x^3 = 28x^3
\][/tex]
[tex]\[
7 \cdot (-3x) = -21x
\][/tex]
[tex]\[
7 \cdot 5 = 35
\][/tex]
Distribute each term of \(-8x^3\):
[tex]\[
-8x^3 \cdot 4x^3 = -32x^6
\][/tex]
[tex]\[
-8x^3 \cdot (-3x) = 24x^4
\][/tex]
[tex]\[
-8x^3 \cdot 5 = -40x^3
\][/tex]
Distribute each term of \(x\):
[tex]\[
x \cdot 4x^3 = 4x^4
\][/tex]
[tex]\[
x \cdot (-3x) = -3x^2
\][/tex]
[tex]\[
x \cdot 5 = 5x
\][/tex]
Now we combine all the results:
[tex]\[
-32x^6 + 24x^4 + 4x^4 + 28x^3 - 40x^3 -3x^2 - 21x + 5x + 35
\][/tex]
Next, combine like terms:
[tex]\[
-32x^6 + (24x^4 + 4x^4) + (28x^3 - 40x^3) - 3x^2 + (-21x + 5x) + 35
\][/tex]
Simplify the coefficients:
[tex]\[
-32x^6 + 28x^4 - 12x^3 - 3x^2 - 16x + 35
\][/tex]
Therefore, the final simplified polynomial is:
[tex]\[
-32x^6 + 28.0x^4 - 12x^3 - 3.0x^2 - 16.0x + 35
\][/tex]
