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]
[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]
