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Which shows one way to determine the factors of [tex]$x^3+4x^2+5x+20$[/tex] by grouping?

A. [tex]x(x^2+4)+5(x^2+4)[/tex]
B. [tex]x^2(x+4)+5(x+4)[/tex]
C. [tex]x^2(x+5)+4(x+5)[/tex]
D. [tex]x(x^2+5)+4x(x^2+5)[/tex]

Sagot :

GinnyAnsweridn
To determine the factors of the polynomial [tex]\(x^3 + 4x^2 + 5x + 20\)[/tex] by grouping, we will follow these steps:

1. Group the terms into pairs:

[tex]\[ (x^3 + 4x^2) + (5x + 20) \][/tex]

2. Factor out the common factor from each pair:

From the first group [tex]\((x^3 + 4x^2)\)[/tex], the common factor is [tex]\(x^2\)[/tex]:
[tex]\[ x^2(x + 4) \][/tex]

From the second group [tex]\((5x + 20)\)[/tex], the common factor is 5:
[tex]\[ 5(x + 4) \][/tex]

3. Notice the common binomial factor [tex]\((x + 4)\)[/tex]:

We can now factor out the common binomial [tex]\((x + 4)\)[/tex]:
[tex]\[ x^2(x + 4) + 5(x + 4) \][/tex]

Here, we see that both terms contain a common binomial factor [tex]\((x + 4)\)[/tex].

4. Factor out the common binomial factor:

[tex]\[ (x^2 + 5)(x + 4) \][/tex]

Therefore, one way to determine the factors of [tex]\(x^3 + 4x^2 + 5x + 20\)[/tex] by grouping shows that the correct step-by-step solution for factoring by grouping is:

[tex]\[ x^2(x + 5) + 4(x + 5) \][/tex]

This correctly factors the polynomial as indicated by the grouping.
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