To factor the given polynomial [tex]\( 56x^2 - 8x - 7x + 1 \)[/tex] by grouping, let's go through the detailed steps:
### Step 1: Group pairs of terms with common factors
Group the polynomial into two pairs of terms:
[tex]\[ (56x^2 - 8x) + (-7x + 1) \][/tex]
### Step 2: Factor the Greatest Common Factor (GCF) from each group
#### Group 1: [tex]\( 56x^2 - 8x \)[/tex]
Both terms have a common factor of [tex]\( 8x \)[/tex]:
[tex]\[ 56x^2 - 8x = 8x(7x - 1) \][/tex]
#### Group 2: [tex]\( -7x + 1 \)[/tex]
This group can be factored as:
[tex]\[ -7x + 1 = -1(7x - 1) \][/tex]
So, we now have:
[tex]\[ 8x(7x - 1) - 1(7x - 1) \][/tex]
### Step 3: Use the distributive property
Notice that both terms now contain the common binomial factor [tex]\( (7x - 1) \)[/tex]. We can factor out [tex]\( (7x - 1) \)[/tex]:
[tex]\[ 8x(7x - 1) - 1(7x - 1) = (8x - 1)(7x - 1) \][/tex]
Thus, the factorization of the polynomial [tex]\( 56x^2 - 8x - 7x + 1 \)[/tex] is:
[tex]\[
(8x - 1)(7x - 1)
\][/tex]
