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Which shows one way to determine the factors of [tex]$x^3+5x^2-6x-30$[/tex] by grouping?

A. [tex]x\left(x^2-5\right)+6\left(x^2-5\right)[/tex]
B. [tex]x\left(x^2+5\right)-6\left(x^2+5\right)[/tex]
C. [tex]x^2(x-5)+6(x-5)[/tex]
D. [tex]x^2(x+5)-6(x+5)[/tex]

Sagot :

GinnyAnsweridn
To determine the factors of the polynomial [tex]\( x^3 + 5x^2 - 6x - 30 \)[/tex] by grouping, follow these steps:

1. Group the terms: The given polynomial is [tex]\( x^3 + 5x^2 - 6x - 30 \)[/tex]. We want to group terms in pairs that can be factored easily.

[tex]\[ (x^3 + 5x^2) + (-6x - 30) \][/tex]

2. Factor out the common factors in each group:

- For the first group [tex]\( x^3 + 5x^2 \)[/tex], the common factor is [tex]\( x^2 \)[/tex]:

[tex]\[ x^2 (x + 5) \][/tex]

- For the second group [tex]\( -6x - 30 \)[/tex], the common factor is [tex]\(-6\)[/tex]:

[tex]\[ -6(x + 5) \][/tex]

3. Write the expression after factoring out the common factors:

[tex]\[ x^2 (x + 5) - 6 (x + 5) \][/tex]

4. Factor out the common binomial factor [tex]\((x + 5)\)[/tex]:

[tex]\[ (x^2 - 6)(x + 5) \][/tex]

Therefore, the correct way to factor [tex]\( x^3 + 5x^2 - 6x - 30 \)[/tex] by grouping is:
[tex]\[ x^2 (x + 5) - 6 (x + 5) \][/tex]

This corresponds to the option:
[tex]\[ x^2 (x + 5) - 6 (x + 5) \][/tex]

So, the correct choice is:
[tex]\[ x^2 (x + 5) - 6 (x + 5) \][/tex]
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