To factor the expression [tex]\( 15cd - 45c^2d \)[/tex] step-by-step, follow these steps:
1. Identify the common factors: Look at the coefficients and the variables in each term of the expression. The terms are [tex]\( 15cd \)[/tex] and [tex]\( 45c^2d \)[/tex].
2. Find the greatest common factor (GCF): The GCF of the numerical coefficients 15 and 45 is 15. Both terms also include the variables [tex]\( c \)[/tex] and [tex]\( d \)[/tex]. Since the first term has [tex]\( c \)[/tex] and the second term has [tex]\( c^2 \)[/tex] (which is [tex]\( c \cdot c \)[/tex]), the GCF for the variable part is [tex]\( c \)[/tex]. Both terms also have the variable [tex]\( d \)[/tex].
So, the GCF of the entire expression is [tex]\( 15cd \)[/tex].
3. Factor out the GCF: Write [tex]\( 15cd \)[/tex] in front of a parenthesis and divide each term by the GCF:
[tex]\[
15cd \left( \frac{15cd}{15cd} - \frac{45c^2d}{15cd} \right)
\][/tex]
4. Simplify inside the parenthesis:
- [tex]\( \frac{15cd}{15cd} = 1 \)[/tex]
- [tex]\( \frac{45c^2d}{15cd} = 3c \)[/tex]
So, the expression inside the parenthesis becomes:
[tex]\[
1 - 3c
\][/tex]
5. Write the final factored form: Combine the factored out GCF with the simplified expression inside the parenthesis:
[tex]\[
15cd(1 - 3c)
\][/tex]
Therefore, the factored form of the expression [tex]\( 15cd - 45c^2d \)[/tex] is:
[tex]\[
15cd(1 - 3c)
\][/tex]
Among the given options, this matches closest to the correct answer. If we look at the options provided:
- [tex]\( 3c^2d(5-15) \)[/tex]
- [tex]\( 5cd(3-15c^2d) \)[/tex]
- [tex]\( 3cd(5-15c) \)[/tex]
- [tex]\( 5cd(3-45c^2d) \)[/tex]
None of these are correct. The closest correct answer should be 15cd(1 - 3c), which unfortunately is not listed among the provided options.
Conclusion: There is a mistake in the provided choices. The correct factorization is indeed [tex]\( 15cd(1 - 3c) \)[/tex], but there's no corresponding option given.
