To factor out the greatest common factor (GCF) from the polynomial [tex]\( 45x^3 + 81x^5 - 63x^6 + 18 \)[/tex], follow these steps:
1. Identify the GCF of the coefficients in the polynomial terms.
- The coefficients are [tex]\(45\)[/tex], [tex]\(81\)[/tex], [tex]\(63\)[/tex], and [tex]\(18\)[/tex].
- The GCF of these numbers can be found by determining the highest number that divides all of the coefficients.
- The factors of [tex]\(45\)[/tex] are [tex]\(1, 3, 5, 9, 15, 45\)[/tex].
- The factors of [tex]\(81\)[/tex] are [tex]\(1, 3, 9, 27, 81\)[/tex].
- The factors of [tex]\(63\)[/tex] are [tex]\(1, 3, 7, 9, 21, 63\)[/tex].
- The factors of [tex]\(18\)[/tex] are [tex]\(1, 2, 3, 6, 9, 18\)[/tex].
- The largest number common to all sets of factors is [tex]\(9\)[/tex].
2. Include the variable part in the GCF, if necessary.
- The variable terms are [tex]\(x^3\)[/tex], [tex]\(x^5\)[/tex], and [tex]\(x^6\)[/tex]. The constants term [tex]\(18\)[/tex] does not have a variable.
- The GCF of the variable part is the lowest power of [tex]\(x\)[/tex], which is [tex]\(x^3\)[/tex].
- However, since the constant term [tex]\(18\)[/tex] does not include [tex]\(x\)[/tex], the GCF for the variable part is considered [tex]\(1\)[/tex].
3. Factor out the GCF from the polynomial:
- The combined GCF of the entire polynomial is [tex]\(9\)[/tex].
Let's rewrite each term of the polynomial by factoring [tex]\(9\)[/tex] out:
- [tex]\(45x^3 / 9 = 5x^3\)[/tex]
- [tex]\(81x^5 / 9 = 9x^5\)[/tex]
- [tex]\(63x^6 / 9 = 7x^6\)[/tex]
- [tex]\(18 / 9 = 2\)[/tex]
Thus, factoring out the GCF [tex]\(9\)[/tex] from each term, we get:
[tex]\[
45x^3 + 81x^5 - 63x^6 + 18 = 9(5x^3 + 9x^5 - 7x^6 + 2)
\][/tex]
Since there seems to be a mistake in the signs from previous calculations, the correct factored polynomial should indeed be:
[tex]\[
45 x^3 + 81 x^5 - 63 x^6 + 18 = -9(7x^6 - 9x^5 - 5x^3 - 2)
\][/tex]
Therefore, the factored form of the polynomial is:
[tex]\[
-9(7x^6 - 9x^5 - 5x^3 - 2)
\][/tex]
