Certainly! Let's factorize each of the given expressions step-by-step.
### (i) [tex]\( 16xy - 24x^2y^2 - 32xy^2 \)[/tex]
1. Identify the common factors in each term:
All three terms have the factors [tex]\(x\)[/tex], [tex]\(y\)[/tex], and a numerical coefficient.
2. Factor out the greatest common factor (GCF):
- The GCF of the coefficients 16, 24, and 32 is 8.
- From the variables, the least power of [tex]\( x \)[/tex] is [tex]\( x \)[/tex], and the least power of [tex]\( y \)[/tex] is [tex]\( y \)[/tex].
So, the GCF is [tex]\( 8xy \)[/tex].
3. Rewrite the expression using the GCF:
[tex]\[
16xy - 24x^2y^2 - 32xy^2 = 8xy(2) - 8xy(3xy) - 8xy(4y)
\][/tex]
4. Simplify each term inside the parentheses:
[tex]\[
16xy - 24x^2y^2 - 32xy^2 = 8xy(2 - 3xy - 4y)
\][/tex]
Thus, the factorized form of [tex]\( 16xy - 24x^2y^2 - 32xy^2 \)[/tex] is:
[tex]\[
\boxed{-8xy(3xy + 4y - 2)}
\][/tex]
### (iv) [tex]\( 7x^2yz - 21xy^2z + 28xyz^2 \)[/tex]
1. Identify the common factors in each term:
All three terms have the factors [tex]\( x \)[/tex], [tex]\( y \)[/tex], [tex]\( z \)[/tex], and a numerical coefficient.
2. Factor out the greatest common factor (GCF):
- The GCF of the coefficients 7, 21, and 28 is 7.
- From the variables, the least power of [tex]\( x \)[/tex] is [tex]\( x \)[/tex], the least power of [tex]\( y \)[/tex] is [tex]\( y \)[/tex], and the least power of [tex]\( z \)[/tex] is [tex]\( z \)[/tex].
So, the GCF is [tex]\( 7xyz \)[/tex].
3. Rewrite the expression using the GCF:
[tex]\[
7x^2yz - 21xy^2z + 28xyz^2 = 7xyz(x) - 7xyz(3y) + 7xyz(4z)
\][/tex]
4. Simplify each term inside the parentheses:
[tex]\[
7x^2yz - 21xy^2z + 28xyz^2 = 7xyz(x - 3y + 4z)
\][/tex]
Thus, the factorized form of [tex]\( 7x^2yz - 21xy^2z + 28xyz^2 \)[/tex] is:
[tex]\[
\boxed{7xyz(x - 3y + 4z)}
\][/tex]
These are the factorized forms of the given expressions.
