Answer :
We wish to express
[tex]$$
3x^7+12x^5+9x^2 = \text{(missing factor)}\cdot x\Bigl(x+x^3+3\Bigr)
$$[/tex]
by finding the polynomial factor (denoted by [tex]$Q(x)$[/tex]) such that
[tex]$$
3x^7+12x^5+9x^2 = Q(x) \cdot x\Bigl(x+x^3+3\Bigr) + R(x),
$$[/tex]
where [tex]$R(x)$[/tex] is the remainder. Note that in this context the “missing factor” refers to the quotient from dividing the given polynomial by
[tex]$$
x\Bigl(x+x^3+3\Bigr).
$$[/tex]
The first step is to rewrite the divisor in an expanded form. We have
[tex]$$
x\Bigl(x+x^3+3\Bigr)= x\cdot x + x\cdot x^3 + x\cdot 3 = x^2 + x^4+3x.
$$[/tex]
It is more convenient to arrange the terms in descending order:
[tex]$$
D(x)= x^4 + x^2 + 3x.
$$[/tex]
Also, write the original polynomial with all degrees accounted for:
[tex]$$
P(x)=3x^7+0x^6+12x^5+0x^4+0x^3+9x^2+0x+0.
$$[/tex]
Now, we perform polynomial long division of [tex]$P(x)$[/tex] by [tex]$D(x)$[/tex].
1. Divide the leading terms:
The leading term of [tex]$P(x)$[/tex] is [tex]$3x^7$[/tex] and that of [tex]$D(x)$[/tex] is [tex]$x^4$[/tex]. Dividing:
[tex]$$
\frac{3x^7}{x^4} = 3x^3.
$$[/tex]
Thus, the first term of [tex]$Q(x)$[/tex] is [tex]$3x^3$[/tex].
2. Multiply and subtract:
Multiply the entire divisor by [tex]$3x^3$[/tex]:
[tex]$$
3x^3\cdot (x^4+x^2+3x) = 3x^7+3x^5+9x^4.
$$[/tex]
Subtract this product from [tex]$P(x)$[/tex]:
[tex]$$
\begin{aligned}
P(x) - 3x^3\cdot D(x)
&= \Bigl(3x^7+12x^5+9x^2\Bigr) - \Bigl(3x^7+3x^5+9x^4\Bigr)\\[1mm]
&= \bigl(12x^5-3x^5\bigr) - 9x^4 + 9x^2\\[1mm]
&= 9x^5 - 9x^4 + 9x^2.
\end{aligned}
$$[/tex]
3. Next division step:
Now the leading term is [tex]$9x^5$[/tex]. Divide it by [tex]$x^4$[/tex]:
[tex]$$
\frac{9x^5}{x^4} = 9x.
$$[/tex]
So, the next term in [tex]$Q(x)$[/tex] is [tex]$9x$[/tex].
4. Multiply and subtract again:
Multiply the divisor by [tex]$9x$[/tex]:
[tex]$$
9x\cdot (x^4+x^2+3x) = 9x^5+9x^3+27x^2.
$$[/tex]
Subtract this from the current remainder:
[tex]$$
\begin{aligned}
(9x^5 - 9x^4 + 9x^2) - (9x^5+9x^3+27x^2)
&= -9x^4 - 9x^3 - 18x^2.
\end{aligned}
$$[/tex]
5. Next division step:
Now the leading term is [tex]$-9x^4$[/tex]. Divide this by [tex]$x^4$[/tex]:
[tex]$$
\frac{-9x^4}{x^4} = -9.
$$[/tex]
So the next term in [tex]$Q(x)$[/tex] is [tex]$-9$[/tex].
6. Multiply and subtract for the last time:
Multiply the divisor by [tex]$-9$[/tex]:
[tex]$$
-9\cdot (x^4+x^2+3x)= -9x^4 - 9x^2 - 27x.
$$[/tex]
Subtract this from the current remainder:
[tex]$$
\begin{aligned}
(-9x^4 - 9x^3 - 18x^2) - \bigl(-9x^4 - 9x^2 - 27x\bigr)
&= (-9x^4 - 9x^3 - 18x^2) + 9x^4 + 9x^2+27x\\[1mm]
&= -9x^3 - 9x^2 + 27x.
\end{aligned}
$$[/tex]
Since the degree of the new remainder,
[tex]$$
R(x)= -9x^3-9x^2+27x,
$$[/tex]
is lower than the degree of the divisor [tex]$(4)$[/tex], the division process is complete.
Thus, the quotient and remainder are:
[tex]$$
Q(x)= 3x^3+9x-9 \quad\text{and}\quad R(x)= -9x^3-9x^2+27x.
$$[/tex]
Therefore, we have written the polynomial as
[tex]$$
3x^7+12x^5+9x^2 = \Bigl(3x^3+9x-9\Bigr)x\Bigl(x+x^3+3\Bigr) + \Bigl(-9x^3-9x^2+27x\Bigr).
$$[/tex]
In the required form, the missing factor is
[tex]$$
\boxed{3x^3+9x-9}.
$$[/tex]
[tex]$$
3x^7+12x^5+9x^2 = \text{(missing factor)}\cdot x\Bigl(x+x^3+3\Bigr)
$$[/tex]
by finding the polynomial factor (denoted by [tex]$Q(x)$[/tex]) such that
[tex]$$
3x^7+12x^5+9x^2 = Q(x) \cdot x\Bigl(x+x^3+3\Bigr) + R(x),
$$[/tex]
where [tex]$R(x)$[/tex] is the remainder. Note that in this context the “missing factor” refers to the quotient from dividing the given polynomial by
[tex]$$
x\Bigl(x+x^3+3\Bigr).
$$[/tex]
The first step is to rewrite the divisor in an expanded form. We have
[tex]$$
x\Bigl(x+x^3+3\Bigr)= x\cdot x + x\cdot x^3 + x\cdot 3 = x^2 + x^4+3x.
$$[/tex]
It is more convenient to arrange the terms in descending order:
[tex]$$
D(x)= x^4 + x^2 + 3x.
$$[/tex]
Also, write the original polynomial with all degrees accounted for:
[tex]$$
P(x)=3x^7+0x^6+12x^5+0x^4+0x^3+9x^2+0x+0.
$$[/tex]
Now, we perform polynomial long division of [tex]$P(x)$[/tex] by [tex]$D(x)$[/tex].
1. Divide the leading terms:
The leading term of [tex]$P(x)$[/tex] is [tex]$3x^7$[/tex] and that of [tex]$D(x)$[/tex] is [tex]$x^4$[/tex]. Dividing:
[tex]$$
\frac{3x^7}{x^4} = 3x^3.
$$[/tex]
Thus, the first term of [tex]$Q(x)$[/tex] is [tex]$3x^3$[/tex].
2. Multiply and subtract:
Multiply the entire divisor by [tex]$3x^3$[/tex]:
[tex]$$
3x^3\cdot (x^4+x^2+3x) = 3x^7+3x^5+9x^4.
$$[/tex]
Subtract this product from [tex]$P(x)$[/tex]:
[tex]$$
\begin{aligned}
P(x) - 3x^3\cdot D(x)
&= \Bigl(3x^7+12x^5+9x^2\Bigr) - \Bigl(3x^7+3x^5+9x^4\Bigr)\\[1mm]
&= \bigl(12x^5-3x^5\bigr) - 9x^4 + 9x^2\\[1mm]
&= 9x^5 - 9x^4 + 9x^2.
\end{aligned}
$$[/tex]
3. Next division step:
Now the leading term is [tex]$9x^5$[/tex]. Divide it by [tex]$x^4$[/tex]:
[tex]$$
\frac{9x^5}{x^4} = 9x.
$$[/tex]
So, the next term in [tex]$Q(x)$[/tex] is [tex]$9x$[/tex].
4. Multiply and subtract again:
Multiply the divisor by [tex]$9x$[/tex]:
[tex]$$
9x\cdot (x^4+x^2+3x) = 9x^5+9x^3+27x^2.
$$[/tex]
Subtract this from the current remainder:
[tex]$$
\begin{aligned}
(9x^5 - 9x^4 + 9x^2) - (9x^5+9x^3+27x^2)
&= -9x^4 - 9x^3 - 18x^2.
\end{aligned}
$$[/tex]
5. Next division step:
Now the leading term is [tex]$-9x^4$[/tex]. Divide this by [tex]$x^4$[/tex]:
[tex]$$
\frac{-9x^4}{x^4} = -9.
$$[/tex]
So the next term in [tex]$Q(x)$[/tex] is [tex]$-9$[/tex].
6. Multiply and subtract for the last time:
Multiply the divisor by [tex]$-9$[/tex]:
[tex]$$
-9\cdot (x^4+x^2+3x)= -9x^4 - 9x^2 - 27x.
$$[/tex]
Subtract this from the current remainder:
[tex]$$
\begin{aligned}
(-9x^4 - 9x^3 - 18x^2) - \bigl(-9x^4 - 9x^2 - 27x\bigr)
&= (-9x^4 - 9x^3 - 18x^2) + 9x^4 + 9x^2+27x\\[1mm]
&= -9x^3 - 9x^2 + 27x.
\end{aligned}
$$[/tex]
Since the degree of the new remainder,
[tex]$$
R(x)= -9x^3-9x^2+27x,
$$[/tex]
is lower than the degree of the divisor [tex]$(4)$[/tex], the division process is complete.
Thus, the quotient and remainder are:
[tex]$$
Q(x)= 3x^3+9x-9 \quad\text{and}\quad R(x)= -9x^3-9x^2+27x.
$$[/tex]
Therefore, we have written the polynomial as
[tex]$$
3x^7+12x^5+9x^2 = \Bigl(3x^3+9x-9\Bigr)x\Bigl(x+x^3+3\Bigr) + \Bigl(-9x^3-9x^2+27x\Bigr).
$$[/tex]
In the required form, the missing factor is
[tex]$$
\boxed{3x^3+9x-9}.
$$[/tex]
