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Answer :
To factor the polynomial
[tex]$$
7x^7 + 6x^6 + 3x^5,
$$[/tex]
we start by looking for a common factor in all the terms.
1. Notice that each term contains at least [tex]$x^5$[/tex]. That is,
[tex]$$
7x^7 = x^5 \cdot 7x^2,\quad 6x^6 = x^5 \cdot 6x,\quad 3x^5 = x^5 \cdot 3.
$$[/tex]
2. Factor out the common factor [tex]$x^5$[/tex] from the entire expression:
[tex]$$
7x^7 + 6x^6 + 3x^5 = x^5(7x^2 + 6x + 3).
$$[/tex]
3. Next, examine the quadratic expression [tex]$7x^2 + 6x + 3$[/tex]. We can check if it factors further over the integers. To do so, calculate the discriminant:
[tex]$$
D = b^2 - 4ac = 6^2 - 4(7)(3) = 36 - 84 = -48.
$$[/tex]
Since the discriminant is negative, the quadratic does not factor into real linear factors over the integers.
Thus, the fully factored form of the polynomial is
[tex]$$
\boxed{x^5(7x^2+6x+3)}.
$$[/tex]
[tex]$$
7x^7 + 6x^6 + 3x^5,
$$[/tex]
we start by looking for a common factor in all the terms.
1. Notice that each term contains at least [tex]$x^5$[/tex]. That is,
[tex]$$
7x^7 = x^5 \cdot 7x^2,\quad 6x^6 = x^5 \cdot 6x,\quad 3x^5 = x^5 \cdot 3.
$$[/tex]
2. Factor out the common factor [tex]$x^5$[/tex] from the entire expression:
[tex]$$
7x^7 + 6x^6 + 3x^5 = x^5(7x^2 + 6x + 3).
$$[/tex]
3. Next, examine the quadratic expression [tex]$7x^2 + 6x + 3$[/tex]. We can check if it factors further over the integers. To do so, calculate the discriminant:
[tex]$$
D = b^2 - 4ac = 6^2 - 4(7)(3) = 36 - 84 = -48.
$$[/tex]
Since the discriminant is negative, the quadratic does not factor into real linear factors over the integers.
Thus, the fully factored form of the polynomial is
[tex]$$
\boxed{x^5(7x^2+6x+3)}.
$$[/tex]
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