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Answer :
Sure! Let's break down the expression [tex]\(-33x^5 - 3x^4 + 12x^3 - 21x^2\)[/tex] and factor it to simplify.
1. Identify the Common Factor:
- Look at all the terms:
[tex]\(-33x^5\)[/tex], [tex]\(-3x^4\)[/tex], [tex]\(12x^3\)[/tex], and [tex]\(-21x^2\)[/tex].
- Each term has at least an [tex]\(x^2\)[/tex] factor.
2. Factor Out the Common Factor:
- You can factor [tex]\(x^2\)[/tex] out from each of the terms:
[tex]\[
-33x^5 - 3x^4 + 12x^3 - 21x^2 = x^2(-33x^3 - 3x^2 + 12x - 21)
\][/tex]
3. Factor the Remaining Polynomial:
- Now, you need to factor the remaining polynomial inside the parentheses.
- The expression inside is [tex]\(-33x^3 - 3x^2 + 12x - 21\)[/tex].
- Notice that each coefficient can be factored by 3:
[tex]\[
-33x^3 - 3x^2 + 12x - 21 = 3(-11x^3 - x^2 + 4x - 7)
\][/tex]
4. Combine Factored Parts:
- Incorporate the factor of 3 into the expression:
[tex]\[
x^2 \cdot 3(-11x^3 - x^2 + 4x - 7) = 3x^2(-11x^3 - x^2 + 4x - 7)
\][/tex]
So, the simplified factored form of the expression [tex]\(-33x^5 - 3x^4 + 12x^3 - 21x^2\)[/tex] is:
[tex]\[
3x^2(-11x^3 - x^2 + 4x - 7)
\][/tex]
This is the factored version of your original expression, showing all the steps used to break it down into simpler parts.
1. Identify the Common Factor:
- Look at all the terms:
[tex]\(-33x^5\)[/tex], [tex]\(-3x^4\)[/tex], [tex]\(12x^3\)[/tex], and [tex]\(-21x^2\)[/tex].
- Each term has at least an [tex]\(x^2\)[/tex] factor.
2. Factor Out the Common Factor:
- You can factor [tex]\(x^2\)[/tex] out from each of the terms:
[tex]\[
-33x^5 - 3x^4 + 12x^3 - 21x^2 = x^2(-33x^3 - 3x^2 + 12x - 21)
\][/tex]
3. Factor the Remaining Polynomial:
- Now, you need to factor the remaining polynomial inside the parentheses.
- The expression inside is [tex]\(-33x^3 - 3x^2 + 12x - 21\)[/tex].
- Notice that each coefficient can be factored by 3:
[tex]\[
-33x^3 - 3x^2 + 12x - 21 = 3(-11x^3 - x^2 + 4x - 7)
\][/tex]
4. Combine Factored Parts:
- Incorporate the factor of 3 into the expression:
[tex]\[
x^2 \cdot 3(-11x^3 - x^2 + 4x - 7) = 3x^2(-11x^3 - x^2 + 4x - 7)
\][/tex]
So, the simplified factored form of the expression [tex]\(-33x^5 - 3x^4 + 12x^3 - 21x^2\)[/tex] is:
[tex]\[
3x^2(-11x^3 - x^2 + 4x - 7)
\][/tex]
This is the factored version of your original expression, showing all the steps used to break it down into simpler parts.
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