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Answer :
To factor out the greatest common factor (GCF) from the expression [tex]\(20x^3y^4 - 12x^2y^3\)[/tex], follow these steps:
1. Identify the GCF of the coefficients:
- The coefficients are 20 and 12.
- The prime factorization of 20 is [tex]\(2^2 \times 5\)[/tex].
- The prime factorization of 12 is [tex]\(2^2 \times 3\)[/tex].
- The GCF of 20 and 12 is the highest power of common prime factors. Here, it is [tex]\(2^2 = 4\)[/tex].
2. Identify the GCF of the variable parts:
- For [tex]\(x\)[/tex]:
- The powers of [tex]\(x\)[/tex] are 3 and 2.
- The GCF of [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex] is [tex]\(x^2\)[/tex] (the smallest power of [tex]\(x\)[/tex]).
- For [tex]\(y\)[/tex]:
- The powers of [tex]\(y\)[/tex] are 4 and 3.
- The GCF of [tex]\(y^4\)[/tex] and [tex]\(y^3\)[/tex] is [tex]\(y^3\)[/tex] (the smallest power of [tex]\(y\)[/tex]).
3. Combine the GCFs:
- The overall GCF of the expression is [tex]\(4x^2y^3\)[/tex].
4. Factor out the GCF:
- Rewrite each term by factoring the GCF out:
- [tex]\(20x^3y^4 = 4x^2y^3 \times 5xy\)[/tex]
- [tex]\(12x^2y^3 = 4x^2y^3 \times 3\)[/tex]
- Substitute these factored forms back into the expression:
[tex]\[
20x^3y^4 - 12x^2y^3 = 4x^2y^3(5xy) - 4x^2y^3(3)
\][/tex]
5. Express the expression as a product:
- Factor the GCF from the entire expression:
[tex]\[
20x^3y^4 - 12x^2y^3 = 4x^2y^3(5xy - 3)
\][/tex]
So, the factored form of the expression is [tex]\(4x^2y^3(5xy - 3)\)[/tex].
1. Identify the GCF of the coefficients:
- The coefficients are 20 and 12.
- The prime factorization of 20 is [tex]\(2^2 \times 5\)[/tex].
- The prime factorization of 12 is [tex]\(2^2 \times 3\)[/tex].
- The GCF of 20 and 12 is the highest power of common prime factors. Here, it is [tex]\(2^2 = 4\)[/tex].
2. Identify the GCF of the variable parts:
- For [tex]\(x\)[/tex]:
- The powers of [tex]\(x\)[/tex] are 3 and 2.
- The GCF of [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex] is [tex]\(x^2\)[/tex] (the smallest power of [tex]\(x\)[/tex]).
- For [tex]\(y\)[/tex]:
- The powers of [tex]\(y\)[/tex] are 4 and 3.
- The GCF of [tex]\(y^4\)[/tex] and [tex]\(y^3\)[/tex] is [tex]\(y^3\)[/tex] (the smallest power of [tex]\(y\)[/tex]).
3. Combine the GCFs:
- The overall GCF of the expression is [tex]\(4x^2y^3\)[/tex].
4. Factor out the GCF:
- Rewrite each term by factoring the GCF out:
- [tex]\(20x^3y^4 = 4x^2y^3 \times 5xy\)[/tex]
- [tex]\(12x^2y^3 = 4x^2y^3 \times 3\)[/tex]
- Substitute these factored forms back into the expression:
[tex]\[
20x^3y^4 - 12x^2y^3 = 4x^2y^3(5xy) - 4x^2y^3(3)
\][/tex]
5. Express the expression as a product:
- Factor the GCF from the entire expression:
[tex]\[
20x^3y^4 - 12x^2y^3 = 4x^2y^3(5xy - 3)
\][/tex]
So, the factored form of the expression is [tex]\(4x^2y^3(5xy - 3)\)[/tex].
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