Answer :
To factor the greatest common factor (GCF) out of the polynomial [tex]\(12x^6 + 4x^4 + 20x^3\)[/tex], we can follow these steps:
1. Identify the coefficients of the terms: The polynomial has the terms [tex]\(12x^6\)[/tex], [tex]\(4x^4\)[/tex], and [tex]\(20x^3\)[/tex]. The coefficients are 12, 4, and 20.
2. Determine the GCF of the coefficients:
- List the factors of each coefficient:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 4: 1, 2, 4
- Factors of 20: 1, 2, 4, 5, 10, 20
- The greatest common factor among these numbers is 4.
3. Determine the smallest power of [tex]\(x\)[/tex] in all terms:
- The variable part in each term has powers [tex]\(x^6\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex].
- The smallest power of [tex]\(x\)[/tex] among these is [tex]\(x^3\)[/tex].
4. The GCF of the polynomial:
- Combine the GCF of the coefficients and the smallest power of [tex]\(x\)[/tex]: [tex]\(4x^3\)[/tex].
5. Factor the GCF out of each term:
- Divide each term by the GCF:
- [tex]\( \frac{12x^6}{4x^3} = 3x^3 \)[/tex]
- [tex]\( \frac{4x^4}{4x^3} = x \)[/tex]
- [tex]\( \frac{20x^3}{4x^3} = 5 \)[/tex]
6. Write the factored form:
- The polynomial can now be expressed as the product of the GCF and the simplified terms:
- [tex]\( 12x^6 + 4x^4 + 20x^3 = 4x^3(3x^3 + x + 5) \)[/tex]
So, the polynomial [tex]\(12x^6 + 4x^4 + 20x^3\)[/tex] is factored as [tex]\(4x^3(3x^3 + x + 5)\)[/tex].
1. Identify the coefficients of the terms: The polynomial has the terms [tex]\(12x^6\)[/tex], [tex]\(4x^4\)[/tex], and [tex]\(20x^3\)[/tex]. The coefficients are 12, 4, and 20.
2. Determine the GCF of the coefficients:
- List the factors of each coefficient:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 4: 1, 2, 4
- Factors of 20: 1, 2, 4, 5, 10, 20
- The greatest common factor among these numbers is 4.
3. Determine the smallest power of [tex]\(x\)[/tex] in all terms:
- The variable part in each term has powers [tex]\(x^6\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex].
- The smallest power of [tex]\(x\)[/tex] among these is [tex]\(x^3\)[/tex].
4. The GCF of the polynomial:
- Combine the GCF of the coefficients and the smallest power of [tex]\(x\)[/tex]: [tex]\(4x^3\)[/tex].
5. Factor the GCF out of each term:
- Divide each term by the GCF:
- [tex]\( \frac{12x^6}{4x^3} = 3x^3 \)[/tex]
- [tex]\( \frac{4x^4}{4x^3} = x \)[/tex]
- [tex]\( \frac{20x^3}{4x^3} = 5 \)[/tex]
6. Write the factored form:
- The polynomial can now be expressed as the product of the GCF and the simplified terms:
- [tex]\( 12x^6 + 4x^4 + 20x^3 = 4x^3(3x^3 + x + 5) \)[/tex]
So, the polynomial [tex]\(12x^6 + 4x^4 + 20x^3\)[/tex] is factored as [tex]\(4x^3(3x^3 + x + 5)\)[/tex].
