To factor the polynomial [tex]\( f(x) = 6x^2 - 25x + 21 \)[/tex], we will find its factors step-by-step.
### Step-by-Step Solution:
1. Identify the factors of the quadratic term and the constant term:
The polynomial [tex]\( 6x^2 - 25x + 21 \)[/tex] can be factored into two binomials of the form [tex]\((ax + b)(cx + d)\)[/tex].
2. Find pairs of factors for the quadratic term [tex]\( 6x^2 \)[/tex]:
Possible pairs for [tex]\( 6x^2 \)[/tex] are:
[tex]\[
(6x, x), (3x, 2x)
\][/tex]
3. Find pairs of factors for the constant term [tex]\( 21 \)[/tex]:
Possible pairs for [tex]\( 21 \)[/tex] are:
[tex]\[
(1, 21), (3, 7), (7, 3), (21, 1)
\][/tex]
4. Form products and find the correct middle term:
We need to find the correct pair that, when used in the binomials, will give us the middle term [tex]\(-25x\)[/tex].
5. Combine the pairs and ensure the middle term matches [tex]\(-25x\)[/tex]:
After testing different combinations to ensure that the middle term obtained from the product is [tex]\(-25x\)[/tex], we find:
[tex]\[
(x - 3)(6x - 7)
\][/tex]
Therefore, the factorization of [tex]\( 6x^2 - 25x + 21 \)[/tex] is correctly found to be:
[tex]\[
(6x - 7)(x - 3)
\][/tex]
### Conclusion:
The polynomial [tex]\( 6x^2 - 25x + 21 \)[/tex] factors to:
[tex]\[
(6x - 7)(x - 3)
\][/tex]
So, the detailed factored form is:
[tex]\[
f(x) = (x - 3)(6x - 7)
\][/tex]
