Factor completely or if prime state this.
12b^2-13bv-35v^2
(3b-7v)(4b+5v)
Thanks
To factor the expression completely, follow these steps:
Step 1: Look for common factors, if any.
In this case, there are no common factors among the terms.
Step 2: We need to find two numbers that multiply to give the constant term (-35v^2) but add up to give the coefficient of the middle term (-13bv).
The factors of -35 are (+7 and -5) or (-7 and +5), and the factors of 12 are (+4 and +3) or (-4 and -3).
Now try combining these factor pairs to check if any of them add up to -13bv.
Combination 1: (+7bv - 5bv)
Combination 2: (-7bv + 5bv)
Combination 3: (+7bv + 5bv)
Combination 4: (-7bv - 5bv)
Out of these combinations, the combination (-7bv + 5bv) adds up to -2bv when simplified.
Step 3: Rewrite the middle term (-13bv) using the factor pair that adds up to it.
12b^2 - 13bv - 35v^2 = 12b^2 - 7bv + 5bv - 35v^2
Step 4: Group the terms and factor by grouping.
(12b^2 - 7bv) + (5bv - 35v^2)
Step 5: Factor out the greatest common factor (GCF) from each grouped term.
5bv is the common factor in the first group, and 5v is the common factor in the second group.
5bv(2b - 7) + 5v(2b - 7)
Step 6: Notice that (2b - 7) is common to both terms. Factor it out.
(2b - 7)(5bv + 5v)
So, the factored form of the expression 12b^2 - 13bv - 35v^2 is (2b - 7)(5bv + 5v).
If you want to check if the factors are prime, you can further simplify the expression:
(2b - 7)(5bv + 5v) = (2b - 7)(5v(b + 1))
Therefore, the expression is fully factored and not prime.