Answer :
We wish to express
[tex]$$
3x^7+12x^5+9x^2
$$[/tex]
in the form
[tex]$$
Q(x)\cdot x\Bigl(x+x^3+3\Bigr),
$$[/tex]
which can be rewritten as
[tex]$$
3x^7+12x^5+9x^2 = Q(x) \cdot \Bigl(x^4+x^2+3x\Bigr).
$$[/tex]
Our goal is to find the polynomial [tex]$Q(x)$[/tex] (and, if necessary, a remainder) such that
[tex]$$
3x^7+12x^5+9x^2 = Q(x) \cdot \Bigl(x^4+x^2+3x\Bigr) + R(x),
$$[/tex]
with the degree of [tex]$R(x)$[/tex] less than the degree of the divisor [tex]$x^4+x^2+3x$[/tex].
We perform polynomial division step by step:
1. Notice that the divisor is
[tex]$$
x(x+x^3+3)= x^4+x^2+3x.
$$[/tex]
2. Divide the leading term of the dividend, [tex]$3x^7$[/tex], by the leading term of the divisor, [tex]$x^4$[/tex], to obtain the first term of the quotient:
[tex]$$
\frac{3x^7}{x^4} = 3x^3.
$$[/tex]
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 from the original dividend:
[tex]$$
(3x^7+12x^5+9x^2) - (3x^7+3x^5+9x^4)= 9x^5-9x^4+9x^2.
$$[/tex]
3. Now, divide the new leading term [tex]$9x^5$[/tex] by [tex]$x^4$[/tex] to obtain the next term of the quotient:
[tex]$$
\frac{9x^5}{x^4} = 9x.
$$[/tex]
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 polynomial:
[tex]$$
(9x^5-9x^4+9x^2) - (9x^5+9x^3+27x^2)= -9x^4-9x^3-18x^2.
$$[/tex]
4. Finally, divide the leading term [tex]$-9x^4$[/tex] by [tex]$x^4$[/tex] to obtain the next term:
[tex]$$
\frac{-9x^4}{x^4} = -9.
$$[/tex]
Multiply the divisor by [tex]$-9$[/tex]:
[tex]$$
-9\cdot (x^4+x^2+3x)= -9x^4-9x^2-27x.
$$[/tex]
Subtract this product from the current polynomial:
[tex]$$
(-9x^4-9x^3-18x^2) - (-9x^4-9x^2-27x)= -9x^3-9x^2+27x.
$$[/tex]
After these steps, the quotient obtained is
[tex]$$
Q(x)=3x^3+9x-9,
$$[/tex]
and the remainder is
[tex]$$
R(x)=-9x^3-9x^2+27x.
$$[/tex]
Thus, the complete division gives:
[tex]$$
3x^7+12x^5+9x^2 = \Bigl(3x^3+9x-9\Bigr)\Bigl(x^4+x^2+3x\Bigr) + \Bigl(-9x^3-9x^2+27x\Bigr).
$$[/tex]
Since our goal was to write [tex]$3x^7+12x^5+9x^2$[/tex] in the form
[tex]$$
\text{[?]}\cdot x\Bigl(x+x^3+3\Bigr),
$$[/tex]
the missing factor in the bracket is
[tex]$$
3x^3+9x-9.
$$[/tex]
This is our final answer.
[tex]$$
3x^7+12x^5+9x^2
$$[/tex]
in the form
[tex]$$
Q(x)\cdot x\Bigl(x+x^3+3\Bigr),
$$[/tex]
which can be rewritten as
[tex]$$
3x^7+12x^5+9x^2 = Q(x) \cdot \Bigl(x^4+x^2+3x\Bigr).
$$[/tex]
Our goal is to find the polynomial [tex]$Q(x)$[/tex] (and, if necessary, a remainder) such that
[tex]$$
3x^7+12x^5+9x^2 = Q(x) \cdot \Bigl(x^4+x^2+3x\Bigr) + R(x),
$$[/tex]
with the degree of [tex]$R(x)$[/tex] less than the degree of the divisor [tex]$x^4+x^2+3x$[/tex].
We perform polynomial division step by step:
1. Notice that the divisor is
[tex]$$
x(x+x^3+3)= x^4+x^2+3x.
$$[/tex]
2. Divide the leading term of the dividend, [tex]$3x^7$[/tex], by the leading term of the divisor, [tex]$x^4$[/tex], to obtain the first term of the quotient:
[tex]$$
\frac{3x^7}{x^4} = 3x^3.
$$[/tex]
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 from the original dividend:
[tex]$$
(3x^7+12x^5+9x^2) - (3x^7+3x^5+9x^4)= 9x^5-9x^4+9x^2.
$$[/tex]
3. Now, divide the new leading term [tex]$9x^5$[/tex] by [tex]$x^4$[/tex] to obtain the next term of the quotient:
[tex]$$
\frac{9x^5}{x^4} = 9x.
$$[/tex]
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 polynomial:
[tex]$$
(9x^5-9x^4+9x^2) - (9x^5+9x^3+27x^2)= -9x^4-9x^3-18x^2.
$$[/tex]
4. Finally, divide the leading term [tex]$-9x^4$[/tex] by [tex]$x^4$[/tex] to obtain the next term:
[tex]$$
\frac{-9x^4}{x^4} = -9.
$$[/tex]
Multiply the divisor by [tex]$-9$[/tex]:
[tex]$$
-9\cdot (x^4+x^2+3x)= -9x^4-9x^2-27x.
$$[/tex]
Subtract this product from the current polynomial:
[tex]$$
(-9x^4-9x^3-18x^2) - (-9x^4-9x^2-27x)= -9x^3-9x^2+27x.
$$[/tex]
After these steps, the quotient obtained is
[tex]$$
Q(x)=3x^3+9x-9,
$$[/tex]
and the remainder is
[tex]$$
R(x)=-9x^3-9x^2+27x.
$$[/tex]
Thus, the complete division gives:
[tex]$$
3x^7+12x^5+9x^2 = \Bigl(3x^3+9x-9\Bigr)\Bigl(x^4+x^2+3x\Bigr) + \Bigl(-9x^3-9x^2+27x\Bigr).
$$[/tex]
Since our goal was to write [tex]$3x^7+12x^5+9x^2$[/tex] in the form
[tex]$$
\text{[?]}\cdot x\Bigl(x+x^3+3\Bigr),
$$[/tex]
the missing factor in the bracket is
[tex]$$
3x^3+9x-9.
$$[/tex]
This is our final answer.
