High School

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------------------------------------------------ Factor by grouping:

\[ 4x^5 - 7x^4 + 12x - 21 \]

Answer :

We start with the expression

[tex]$$
4x^5 - 7x^4 + 12x - 21.
$$[/tex]

Step 1. Group the terms.

Group the first two terms and the last two terms:

[tex]$$
(4x^5 - 7x^4) + (12x - 21).
$$[/tex]

Step 2. Factor each group.

In the first group, factor out [tex]$x^4$[/tex]:

[tex]$$
4x^5 - 7x^4 = x^4 (4x - 7).
$$[/tex]

In the second group, factor out [tex]$3$[/tex]:

[tex]$$
12x - 21 = 3 (4x - 7).
$$[/tex]

Step 3. Factor out the common binomial.

Now the expression becomes:

[tex]$$
x^4 (4x - 7) + 3 (4x - 7).
$$[/tex]

Both terms contain the common factor [tex]$(4x - 7)$[/tex], which we factor out:

[tex]$$
(4x - 7)\left(x^4 + 3\right).
$$[/tex]

Thus, the fully factored form of the polynomial is

[tex]$$
(4x - 7)(x^4 + 3).
$$[/tex]