To divide the polynomial [tex]\( a(x) = -14x^7 + 8x^5 - 6x^4 + 5x \)[/tex] by the polynomial [tex]\( b(x) = x^3 \)[/tex], we need to determine the quotient polynomial [tex]\( q(x) \)[/tex] and the remainder polynomial [tex]\( r(x) \)[/tex] such that:
[tex]\[
\frac{a(x)}{b(x)} = q(x) + \frac{r(x)}{b(x)}
\][/tex]
where the degree of [tex]\( r(x) \)[/tex] is less than the degree of [tex]\( b(x) \)[/tex].
Here, the degree of [tex]\( b(x) = x^3 \)[/tex] is 3. Therefore, the degree of the remainder [tex]\( r(x) \)[/tex] must be less than 3.
After performing the polynomial division, we find:
[tex]\[
a(x) = b(x) \cdot q(x) + r(x)
\][/tex]
The quotient [tex]\( q(x) \)[/tex] and the remainder [tex]\( r(x) \)[/tex] are:
[tex]\[
q(x) = 2x(-7x^3 + 4x - 3)
\][/tex]
and
[tex]\[
r(x) = 5x
\][/tex]
So, the quotient polynomial [tex]\( q(x) \)[/tex] is:
[tex]\[
q(x) = 2x(-7x^3 + 4x - 3)
\][/tex]
And the remainder polynomial [tex]\( r(x) \)[/tex] is:
[tex]\[
r(x) = 5x
\][/tex]
Therefore, the quotient [tex]\( q(x) \)[/tex] is [tex]\( 2x(-7x^3 + 4x - 3) \)[/tex], and the remainder [tex]\( r(x) \)[/tex] is [tex]\( 5x \)[/tex].
