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The product of a binomial and a trinomial is [tex]$x^3 + 3x^2 - x + 2x^2 + 6x - 2$[/tex]. Which expression is equivalent to this product after it has been fully simplified?

A. [tex]$x^3 + 5x^2 + 5x - 2$[/tex]
B. [tex][tex]$x^3 + 2x^2 + 8x - 2$[/tex][/tex]
C. [tex]$x^3 + 11x^2 - 2$[/tex]
D. [tex]$x^3 + 10x^2 - 2$[/tex]

Sagot :

GinnyAnsweridn
To simplify the given polynomial [tex]\(x^3 + 3x^2 - x + 2x^2 + 6x - 2\)[/tex], follow these steps:

1. Group the like terms together:
- [tex]\(x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(3x^2 + 2x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-x + 6x\)[/tex]
- Constant term: [tex]\(-2\)[/tex]

2. Combine the like terms:
- The [tex]\(x^3\)[/tex] term remains [tex]\(x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(3x^2 + 2x^2 = 5x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-x + 6x = 5x\)[/tex].
- The constant term remains [tex]\(-2\)[/tex].

3. Write out the simplified polynomial:
- [tex]\(x^3 + 5x^2 + 5x - 2\)[/tex]

Thus, the equivalent expression to the polynomial [tex]\(x^3 + 3x^2 - x + 2x^2 + 6x - 2\)[/tex] after full simplification is [tex]\(x^3 + 5x^2 + 5x - 2\)[/tex].

From the given options, this matches [tex]\(x^3 + 5x^2 + 5x - 2\)[/tex].
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