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Answer :
To find the greatest common factor (GCF) of the expressions [tex]\(27 x^4\)[/tex], [tex]\(63 x^5\)[/tex], and [tex]\(45 x^3\)[/tex], we will follow these steps:
1. Identify the coefficients and the powers of [tex]\(x\)[/tex]:
- For [tex]\(27 x^4\)[/tex], the coefficient is 27 and the power of [tex]\(x\)[/tex] is 4.
- For [tex]\(63 x^5\)[/tex], the coefficient is 63 and the power of [tex]\(x\)[/tex] is 5.
- For [tex]\(45 x^3\)[/tex], the coefficient is 45 and the power of [tex]\(x\)[/tex] is 3.
2. Find the greatest common divisor (GCD) of the coefficients:
- The coefficients we have are 27, 63, and 45.
- We need to find the largest number that divides all these coefficients without a remainder.
Let's break down the numbers into their prime factors:
- 27 = [tex]\(3^3\)[/tex]
- 63 = [tex]\(3^2 \times 7\)[/tex]
- 45 = [tex]\(3^2 \times 5\)[/tex]
The GCD is found by taking the lowest power of each common prime factor. Here, the common prime factor is [tex]\(3\)[/tex] at the lowest power of [tex]\(2\)[/tex].
Thus, the GCD of 27, 63, and 45 is [tex]\(3^2 = 9\)[/tex].
3. Determine the smallest power of [tex]\(x\)[/tex]:
- The powers of [tex]\(x\)[/tex] are 4, 5, and 3.
- The smallest power of [tex]\(x\)[/tex] is 3.
4. Construct the GCF of the expressions:
- Combine the GCD of the coefficients with the smallest power of [tex]\(x\)[/tex].
- The GCF of the expressions is [tex]\(9x^3\)[/tex].
Thus, the greatest common factor of the expressions [tex]\(27 x^4\)[/tex], [tex]\(63 x^5\)[/tex], and [tex]\(45 x^3\)[/tex] is [tex]\(9x^3\)[/tex].
1. Identify the coefficients and the powers of [tex]\(x\)[/tex]:
- For [tex]\(27 x^4\)[/tex], the coefficient is 27 and the power of [tex]\(x\)[/tex] is 4.
- For [tex]\(63 x^5\)[/tex], the coefficient is 63 and the power of [tex]\(x\)[/tex] is 5.
- For [tex]\(45 x^3\)[/tex], the coefficient is 45 and the power of [tex]\(x\)[/tex] is 3.
2. Find the greatest common divisor (GCD) of the coefficients:
- The coefficients we have are 27, 63, and 45.
- We need to find the largest number that divides all these coefficients without a remainder.
Let's break down the numbers into their prime factors:
- 27 = [tex]\(3^3\)[/tex]
- 63 = [tex]\(3^2 \times 7\)[/tex]
- 45 = [tex]\(3^2 \times 5\)[/tex]
The GCD is found by taking the lowest power of each common prime factor. Here, the common prime factor is [tex]\(3\)[/tex] at the lowest power of [tex]\(2\)[/tex].
Thus, the GCD of 27, 63, and 45 is [tex]\(3^2 = 9\)[/tex].
3. Determine the smallest power of [tex]\(x\)[/tex]:
- The powers of [tex]\(x\)[/tex] are 4, 5, and 3.
- The smallest power of [tex]\(x\)[/tex] is 3.
4. Construct the GCF of the expressions:
- Combine the GCD of the coefficients with the smallest power of [tex]\(x\)[/tex].
- The GCF of the expressions is [tex]\(9x^3\)[/tex].
Thus, the greatest common factor of the expressions [tex]\(27 x^4\)[/tex], [tex]\(63 x^5\)[/tex], and [tex]\(45 x^3\)[/tex] is [tex]\(9x^3\)[/tex].
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