Answer :
Let's factor each polynomial completely:
c. [tex]\( 24 m^3 n^4 + 32 m n p \)[/tex]
1. Find the greatest common factor (GCF) of the coefficients:
- Coefficients are 24 and 32.
- The GCF of 24 and 32 is 8.
2. Find the common variables and their least exponents:
- In this case, the common variable across both terms is [tex]\( m \)[/tex] with the least exponent of 1.
3. Factor out the GCF from the polynomial:
[tex]\[
24 m^3 n^4 + 32 m n p = 8m(3 m^2 n^4 + 4 n p)
\][/tex]
Thus, the completely factored form is [tex]\( 8m(3 m^2 n^4 + 4 n p) \)[/tex].
f. [tex]\( 12 x^4 y^2 z - 42 x^3 y^3 z^2 \)[/tex]
1. Find the greatest common factor (GCF) of the coefficients:
- Coefficients are 12 and 42.
- The GCF of 12 and 42 is 6.
2. Find the common variables and their least exponents:
- Variables common in both terms are [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].
- The least exponents are [tex]\( x^3 \)[/tex], [tex]\( y^2 \)[/tex], and [tex]\( z \)[/tex].
3. Factor out the GCF from the polynomial:
- Divide each term by the GCF:
[tex]\[
12 x^4 y^2 z \div 6 x^3 y^2 z = 2x
\][/tex]
[tex]\[
42 x^3 y^3 z^2 \div 6 x^3 y^2 z = 7yz
\][/tex]
- Factoring these out, we get:
[tex]\[
12 x^4 y^2 z - 42 x^3 y^3 z^2 = 6 x^3 y^2 z(2x - 7yz)
\][/tex]
Therefore, the completely factored form is [tex]\( 6 x^3 y^2 z(2x - 7yz) \)[/tex].
c. [tex]\( 24 m^3 n^4 + 32 m n p \)[/tex]
1. Find the greatest common factor (GCF) of the coefficients:
- Coefficients are 24 and 32.
- The GCF of 24 and 32 is 8.
2. Find the common variables and their least exponents:
- In this case, the common variable across both terms is [tex]\( m \)[/tex] with the least exponent of 1.
3. Factor out the GCF from the polynomial:
[tex]\[
24 m^3 n^4 + 32 m n p = 8m(3 m^2 n^4 + 4 n p)
\][/tex]
Thus, the completely factored form is [tex]\( 8m(3 m^2 n^4 + 4 n p) \)[/tex].
f. [tex]\( 12 x^4 y^2 z - 42 x^3 y^3 z^2 \)[/tex]
1. Find the greatest common factor (GCF) of the coefficients:
- Coefficients are 12 and 42.
- The GCF of 12 and 42 is 6.
2. Find the common variables and their least exponents:
- Variables common in both terms are [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].
- The least exponents are [tex]\( x^3 \)[/tex], [tex]\( y^2 \)[/tex], and [tex]\( z \)[/tex].
3. Factor out the GCF from the polynomial:
- Divide each term by the GCF:
[tex]\[
12 x^4 y^2 z \div 6 x^3 y^2 z = 2x
\][/tex]
[tex]\[
42 x^3 y^3 z^2 \div 6 x^3 y^2 z = 7yz
\][/tex]
- Factoring these out, we get:
[tex]\[
12 x^4 y^2 z - 42 x^3 y^3 z^2 = 6 x^3 y^2 z(2x - 7yz)
\][/tex]
Therefore, the completely factored form is [tex]\( 6 x^3 y^2 z(2x - 7yz) \)[/tex].
