Answer :
To factor the GCF (Greatest Common Factor) out of the expression [tex]\( 45x^5y^7 + 33x^3y^3 + 78x^2y^4 \)[/tex], follow these steps:
1. Identify the GCF of the coefficients:
- The coefficients are 45, 33, and 78.
- The prime factors of these numbers are:
- 45 = 3 [tex]\(\cdot\)[/tex] 3 [tex]\(\cdot\)[/tex] 5
- 33 = 3 [tex]\(\cdot\)[/tex] 11
- 78 = 2 [tex]\(\cdot\)[/tex] 3 [tex]\(\cdot\)[/tex] 13
- The common factor among these coefficients is [tex]\( 3 \)[/tex].
2. Identify the GCF for the variables [tex]\( x \)[/tex]:
- The terms have [tex]\( x^5 \)[/tex], [tex]\( x^3 \)[/tex], and [tex]\( x^2 \)[/tex].
- The smallest power of [tex]\( x \)[/tex] among these is [tex]\( x^2 \)[/tex].
3. Identify the GCF for the variables [tex]\( y \)[/tex]:
- The terms contain [tex]\( y^7 \)[/tex], [tex]\( y^3 \)[/tex], and [tex]\( y^4 \)[/tex].
- The smallest power of [tex]\( y \)[/tex] among these is [tex]\( y^3 \)[/tex].
4. Combine the GCFs:
- The GCF of the whole expression is [tex]\( 3x^2y^3 \)[/tex].
5. Factor the GCF out of the expression:
- Divide each term by [tex]\( 3x^2y^3 \)[/tex].
- [tex]\( 45x^5y^7 \div 3x^2y^3 = 15x^3y^4 \)[/tex]
- [tex]\( 33x^3y^3 \div 3x^2y^3 = 11x \)[/tex]
- [tex]\( 78x^2y^4 \div 3x^2y^3 = 26y \)[/tex]
6. Write the expression in factored form:
- The expression factored with GCF is [tex]\( 3x^2y^3(15x^3y^4 + 11x + 26y) \)[/tex].
So, the factored form of the given expression [tex]\( 45x^5y^7 + 33x^3y^3 + 78x^2y^4 \)[/tex] is:
[tex]\[ 3x^2y^3(15x^3y^4 + 11x + 26y) \][/tex]
1. Identify the GCF of the coefficients:
- The coefficients are 45, 33, and 78.
- The prime factors of these numbers are:
- 45 = 3 [tex]\(\cdot\)[/tex] 3 [tex]\(\cdot\)[/tex] 5
- 33 = 3 [tex]\(\cdot\)[/tex] 11
- 78 = 2 [tex]\(\cdot\)[/tex] 3 [tex]\(\cdot\)[/tex] 13
- The common factor among these coefficients is [tex]\( 3 \)[/tex].
2. Identify the GCF for the variables [tex]\( x \)[/tex]:
- The terms have [tex]\( x^5 \)[/tex], [tex]\( x^3 \)[/tex], and [tex]\( x^2 \)[/tex].
- The smallest power of [tex]\( x \)[/tex] among these is [tex]\( x^2 \)[/tex].
3. Identify the GCF for the variables [tex]\( y \)[/tex]:
- The terms contain [tex]\( y^7 \)[/tex], [tex]\( y^3 \)[/tex], and [tex]\( y^4 \)[/tex].
- The smallest power of [tex]\( y \)[/tex] among these is [tex]\( y^3 \)[/tex].
4. Combine the GCFs:
- The GCF of the whole expression is [tex]\( 3x^2y^3 \)[/tex].
5. Factor the GCF out of the expression:
- Divide each term by [tex]\( 3x^2y^3 \)[/tex].
- [tex]\( 45x^5y^7 \div 3x^2y^3 = 15x^3y^4 \)[/tex]
- [tex]\( 33x^3y^3 \div 3x^2y^3 = 11x \)[/tex]
- [tex]\( 78x^2y^4 \div 3x^2y^3 = 26y \)[/tex]
6. Write the expression in factored form:
- The expression factored with GCF is [tex]\( 3x^2y^3(15x^3y^4 + 11x + 26y) \)[/tex].
So, the factored form of the given expression [tex]\( 45x^5y^7 + 33x^3y^3 + 78x^2y^4 \)[/tex] is:
[tex]\[ 3x^2y^3(15x^3y^4 + 11x + 26y) \][/tex]
