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Answer :
To solve the problem of finding which expression is equal to [tex]\((x-3)\left(2x^2-x+3\right)\)[/tex], we need to use the distributive property (also known as expanding the expression).
Let's break it down step-by-step:
1. Expand the Expression:
We'll distribute each term in the first binomial [tex]\((x - 3)\)[/tex] across the trinomial [tex]\((2x^2 - x + 3)\)[/tex]:
[tex]\[
(x - 3)(2x^2 - x + 3)
\][/tex]
2. Distribute [tex]\(x\)[/tex]:
- Multiply [tex]\(x\)[/tex] by each term in the trinomial:
- [tex]\(x \cdot 2x^2 = 2x^3\)[/tex]
- [tex]\(x \cdot (-x) = -x^2\)[/tex]
- [tex]\(x \cdot 3 = 3x\)[/tex]
After distributing [tex]\(x\)[/tex], we have:
[tex]\[
2x^3 - x^2 + 3x
\][/tex]
3. Distribute [tex]\(-3\)[/tex]:
- Multiply [tex]\(-3\)[/tex] by each term in the trinomial:
- [tex]\(-3 \cdot 2x^2 = -6x^2\)[/tex]
- [tex]\(-3 \cdot (-x) = 3x\)[/tex]
- [tex]\(-3 \cdot 3 = -9\)[/tex]
After distributing [tex]\(-3\)[/tex], we have:
[tex]\[
-6x^2 + 3x - 9
\][/tex]
4. Combine Like Terms:
Now we add the results from the multiplication:
[tex]\[
2x^3 - x^2 + 3x - 6x^2 + 3x - 9
\][/tex]
Combine the like terms:
- [tex]\(2x^3\)[/tex] just stays as is.
- Combine [tex]\(-x^2\)[/tex] and [tex]\(-6x^2\)[/tex] to get [tex]\(-7x^2\)[/tex].
- Combine [tex]\(3x\)[/tex] and [tex]\(3x\)[/tex] to get [tex]\(6x\)[/tex].
This gives us:
[tex]\[
2x^3 - 7x^2 + 6x - 9
\][/tex]
So, the expanded form of the expression [tex]\((x-3)(2x^2-x+3)\)[/tex] is [tex]\(2x^3 - 7x^2 + 6x - 9\)[/tex].
Thus, the correct choice from the options provided is:
[tex]\[
2x^3 - 7x^2 + 6x - 9
\][/tex]
Let's break it down step-by-step:
1. Expand the Expression:
We'll distribute each term in the first binomial [tex]\((x - 3)\)[/tex] across the trinomial [tex]\((2x^2 - x + 3)\)[/tex]:
[tex]\[
(x - 3)(2x^2 - x + 3)
\][/tex]
2. Distribute [tex]\(x\)[/tex]:
- Multiply [tex]\(x\)[/tex] by each term in the trinomial:
- [tex]\(x \cdot 2x^2 = 2x^3\)[/tex]
- [tex]\(x \cdot (-x) = -x^2\)[/tex]
- [tex]\(x \cdot 3 = 3x\)[/tex]
After distributing [tex]\(x\)[/tex], we have:
[tex]\[
2x^3 - x^2 + 3x
\][/tex]
3. Distribute [tex]\(-3\)[/tex]:
- Multiply [tex]\(-3\)[/tex] by each term in the trinomial:
- [tex]\(-3 \cdot 2x^2 = -6x^2\)[/tex]
- [tex]\(-3 \cdot (-x) = 3x\)[/tex]
- [tex]\(-3 \cdot 3 = -9\)[/tex]
After distributing [tex]\(-3\)[/tex], we have:
[tex]\[
-6x^2 + 3x - 9
\][/tex]
4. Combine Like Terms:
Now we add the results from the multiplication:
[tex]\[
2x^3 - x^2 + 3x - 6x^2 + 3x - 9
\][/tex]
Combine the like terms:
- [tex]\(2x^3\)[/tex] just stays as is.
- Combine [tex]\(-x^2\)[/tex] and [tex]\(-6x^2\)[/tex] to get [tex]\(-7x^2\)[/tex].
- Combine [tex]\(3x\)[/tex] and [tex]\(3x\)[/tex] to get [tex]\(6x\)[/tex].
This gives us:
[tex]\[
2x^3 - 7x^2 + 6x - 9
\][/tex]
So, the expanded form of the expression [tex]\((x-3)(2x^2-x+3)\)[/tex] is [tex]\(2x^3 - 7x^2 + 6x - 9\)[/tex].
Thus, the correct choice from the options provided is:
[tex]\[
2x^3 - 7x^2 + 6x - 9
\][/tex]
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