Answer :
To factor out the greatest common factor (GCF) from the polynomial [tex]\(45x^7 + 81x^9 - 63x^5 + 18\)[/tex], we will follow these steps:
1. Identify the GCF of the coefficients: The coefficients of the polynomial are 45, 81, 63, and 18. We need to find the greatest common factor of these numbers.
2. Find the GCF of the coefficients:
- The factors of 45 are 1, 3, 5, 9, 15, 45.
- The factors of 81 are 1, 3, 9, 27, 81.
- The factors of 63 are 1, 3, 7, 9, 21, 63.
- The factors of 18 are 1, 2, 3, 6, 9, 18.
- The common factors among these numbers are 1, 3, and 9. The greatest of these is 9.
3. Factor out the GCF:
- Now, we take out 9 from each term in the polynomial:
[tex]\( 45x^7 = 9 \times 5x^7 \)[/tex]
[tex]\( 81x^9 = 9 \times 9x^9 \)[/tex]
[tex]\( 63x^5 = 9 \times 7x^5 \)[/tex]
[tex]\( 18 = 9 \times 2 \)[/tex]
4. Write the polynomial in factored form:
- By factoring out the GCF 9, we can write the polynomial as:
[tex]\[
9(5x^7 + 9x^9 - 7x^5 + 2)
\][/tex]
Therefore, the polynomial [tex]\(45x^7 + 81x^9 - 63x^5 + 18\)[/tex] can be factored as [tex]\(9(5x^7 + 9x^9 - 7x^5 + 2)\)[/tex].
1. Identify the GCF of the coefficients: The coefficients of the polynomial are 45, 81, 63, and 18. We need to find the greatest common factor of these numbers.
2. Find the GCF of the coefficients:
- The factors of 45 are 1, 3, 5, 9, 15, 45.
- The factors of 81 are 1, 3, 9, 27, 81.
- The factors of 63 are 1, 3, 7, 9, 21, 63.
- The factors of 18 are 1, 2, 3, 6, 9, 18.
- The common factors among these numbers are 1, 3, and 9. The greatest of these is 9.
3. Factor out the GCF:
- Now, we take out 9 from each term in the polynomial:
[tex]\( 45x^7 = 9 \times 5x^7 \)[/tex]
[tex]\( 81x^9 = 9 \times 9x^9 \)[/tex]
[tex]\( 63x^5 = 9 \times 7x^5 \)[/tex]
[tex]\( 18 = 9 \times 2 \)[/tex]
4. Write the polynomial in factored form:
- By factoring out the GCF 9, we can write the polynomial as:
[tex]\[
9(5x^7 + 9x^9 - 7x^5 + 2)
\][/tex]
Therefore, the polynomial [tex]\(45x^7 + 81x^9 - 63x^5 + 18\)[/tex] can be factored as [tex]\(9(5x^7 + 9x^9 - 7x^5 + 2)\)[/tex].
