Answer :
To solve this problem, we need to factor out the greatest common factor (GCF) from the expression [tex]\(18x^6 - 45x^5 + 27x^3\)[/tex] and verify by multiplying back to ensure we retrieve the original expression.
1. Identify the GCF:
- Look at the coefficients: 18, 45, and 27. Find the GCF of these numbers.
- The GCF of 18, 45, and 27 is 9.
- Now look at the variable part. Each term has the factor [tex]\(x^3\)[/tex], so the GCF from the variable part is [tex]\(x^3\)[/tex].
- Therefore, the overall GCF of the expression is [tex]\(9x^3\)[/tex].
2. Factor out the GCF:
- Divide each term in the expression by the GCF [tex]\(9x^3\)[/tex] to simplify:
- [tex]\(\frac{18x^6}{9x^3} = 2x^3\)[/tex]
- [tex]\(\frac{-45x^5}{9x^3} = -5x^2\)[/tex]
- [tex]\(\frac{27x^3}{9x^3} = 3\)[/tex]
- This gives the factored expression: [tex]\(9x^3(2x^3 - 5x^2 + 3)\)[/tex].
3. Verify by multiplication:
- Expand [tex]\(9x^3(2x^3 - 5x^2 + 3)\)[/tex] to verify:
- [tex]\(9x^3 \times 2x^3 = 18x^6\)[/tex]
- [tex]\(9x^3 \times -5x^2 = -45x^5\)[/tex]
- [tex]\(9x^3 \times 3 = 27x^3\)[/tex]
- Adding these results together, we retrieve the original expression: [tex]\(18x^6 - 45x^5 + 27x^3\)[/tex].
Thus, the expression [tex]\(18x^6 - 45x^5 + 27x^3\)[/tex] can be factored as [tex]\(9x^3(2x^3 - 5x^2 + 3)\)[/tex], and the verification confirms that this factorization is correct.
1. Identify the GCF:
- Look at the coefficients: 18, 45, and 27. Find the GCF of these numbers.
- The GCF of 18, 45, and 27 is 9.
- Now look at the variable part. Each term has the factor [tex]\(x^3\)[/tex], so the GCF from the variable part is [tex]\(x^3\)[/tex].
- Therefore, the overall GCF of the expression is [tex]\(9x^3\)[/tex].
2. Factor out the GCF:
- Divide each term in the expression by the GCF [tex]\(9x^3\)[/tex] to simplify:
- [tex]\(\frac{18x^6}{9x^3} = 2x^3\)[/tex]
- [tex]\(\frac{-45x^5}{9x^3} = -5x^2\)[/tex]
- [tex]\(\frac{27x^3}{9x^3} = 3\)[/tex]
- This gives the factored expression: [tex]\(9x^3(2x^3 - 5x^2 + 3)\)[/tex].
3. Verify by multiplication:
- Expand [tex]\(9x^3(2x^3 - 5x^2 + 3)\)[/tex] to verify:
- [tex]\(9x^3 \times 2x^3 = 18x^6\)[/tex]
- [tex]\(9x^3 \times -5x^2 = -45x^5\)[/tex]
- [tex]\(9x^3 \times 3 = 27x^3\)[/tex]
- Adding these results together, we retrieve the original expression: [tex]\(18x^6 - 45x^5 + 27x^3\)[/tex].
Thus, the expression [tex]\(18x^6 - 45x^5 + 27x^3\)[/tex] can be factored as [tex]\(9x^3(2x^3 - 5x^2 + 3)\)[/tex], and the verification confirms that this factorization is correct.
