To factor out the common monomial factor in the expression [tex]\( 50a - 100b + 190c \)[/tex], follow these steps:
1. Identify the coefficients and variables in each term of the expression:
- The coefficient of [tex]\(a\)[/tex] in [tex]\(50a\)[/tex] is [tex]\(50\)[/tex].
- The coefficient of [tex]\(b\)[/tex] in [tex]\(-100b\)[/tex] is [tex]\(-100\)[/tex].
- The coefficient of [tex]\(c\)[/tex] in [tex]\(190c\)[/tex] is [tex]\(190\)[/tex].
2. Find the greatest common divisor (GCD) of the coefficients:
- The numbers we have are [tex]\(50\)[/tex], [tex]\(-100\)[/tex], and [tex]\(190\)[/tex].
- The prime factorization of [tex]\(50\)[/tex] is [tex]\(2 \times 5^2\)[/tex].
- The prime factorization of [tex]\(-100\)[/tex] is [tex]\(2^2 \times 5^2 \times (-1)\)[/tex].
- The prime factorization of [tex]\(190\)[/tex] is [tex]\(2 \times 5 \times 19\)[/tex].
3. Determine the GCD of these numbers:
- From the prime factorizations, the smallest power of common prime factors is [tex]\(2 \times 5 = 10\)[/tex].
4. Factor out the GCD from each term in the expression:
- [tex]\(50a = 10 \times 5a\)[/tex].
- [tex]\(-100b = 10 \times (-10b)\)[/tex].
- [tex]\(190c = 10 \times 19c\)[/tex].
5. Rewrite the expression by factoring out the GCD of [tex]\(10\)[/tex]:
[tex]\[
50a - 100b + 190c = 10 \times (5a) + 10 \times (-10b) + 10 \times (19c)
\][/tex]
6. Factor out the common factor [tex]\(10\)[/tex] from the expression:
[tex]\[
50a - 100b + 190c = 10 (5a - 10b + 19c)
\][/tex]
Thus, the expression [tex]\( 50a - 100b + 190c \)[/tex] can be factored as:
[tex]\[
50a - 100b + 190c = 10(5a - 10b + 19c)
\][/tex]
