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Answer :
Given the polynomial
[tex]$$
-3x^4 + 12x^3 - 21x^2,
$$[/tex]
we can start by identifying any common factors in each term. Noticing that each term has a factor of [tex]$-3x^2$[/tex], we factor it out:
[tex]$$
-3x^4 + 12x^3 - 21x^2 = -3x^2\left(\frac{-3x^4}{-3x^2} + \frac{12x^3}{-3x^2} - \frac{21x^2}{-3x^2}\right).
$$[/tex]
Calculating each term inside the parentheses:
1. For [tex]$-3x^4$[/tex] divided by [tex]$-3x^2$[/tex]:
[tex]$$
\frac{-3x^4}{-3x^2} = x^2,
$$[/tex]
2. For [tex]$12x^3$[/tex] divided by [tex]$-3x^2$[/tex]:
[tex]$$
\frac{12x^3}{-3x^2} = -4x,
$$[/tex]
3. For [tex]$-21x^2$[/tex] divided by [tex]$-3x^2$[/tex]:
[tex]$$
\frac{-21x^2}{-3x^2} = 7.
$$[/tex]
So the simplified expression inside the parentheses is:
[tex]$$
x^2 - 4x + 7.
$$[/tex]
Thus, the complete factorization of the polynomial is:
[tex]$$
-3x^4 + 12x^3 - 21x^2 = -3x^2 \left(x^2 - 4x + 7\right).
$$[/tex]
This is the fully factored form of the given polynomial.
[tex]$$
-3x^4 + 12x^3 - 21x^2,
$$[/tex]
we can start by identifying any common factors in each term. Noticing that each term has a factor of [tex]$-3x^2$[/tex], we factor it out:
[tex]$$
-3x^4 + 12x^3 - 21x^2 = -3x^2\left(\frac{-3x^4}{-3x^2} + \frac{12x^3}{-3x^2} - \frac{21x^2}{-3x^2}\right).
$$[/tex]
Calculating each term inside the parentheses:
1. For [tex]$-3x^4$[/tex] divided by [tex]$-3x^2$[/tex]:
[tex]$$
\frac{-3x^4}{-3x^2} = x^2,
$$[/tex]
2. For [tex]$12x^3$[/tex] divided by [tex]$-3x^2$[/tex]:
[tex]$$
\frac{12x^3}{-3x^2} = -4x,
$$[/tex]
3. For [tex]$-21x^2$[/tex] divided by [tex]$-3x^2$[/tex]:
[tex]$$
\frac{-21x^2}{-3x^2} = 7.
$$[/tex]
So the simplified expression inside the parentheses is:
[tex]$$
x^2 - 4x + 7.
$$[/tex]
Thus, the complete factorization of the polynomial is:
[tex]$$
-3x^4 + 12x^3 - 21x^2 = -3x^2 \left(x^2 - 4x + 7\right).
$$[/tex]
This is the fully factored form of the given polynomial.
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