Answer :
To determine if [tex]\(x + 3\)[/tex] is a factor of the polynomial [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex], we can use the Factor Theorem. This theorem states that [tex]\(x + a\)[/tex] is a factor of a polynomial if and only if substituting [tex]\(-a\)[/tex] into the polynomial gives a result of zero.
For the polynomial [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex], we need to check if [tex]\(x + 3\)[/tex] is a factor. Therefore, we substitute [tex]\(x = -3\)[/tex] into the polynomial. Let's evaluate the polynomial at [tex]\(x = -3\)[/tex]:
1. Start with the polynomial: [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex].
2. Substitute [tex]\(x = -3\)[/tex] into the polynomial:
[tex]\[
7(-3)^3 + 27(-3)^2 + 9(-3) - 27
\][/tex]
3. Calculate each term:
- [tex]\((-3)^3 = -27\)[/tex], so [tex]\(7(-27) = -189\)[/tex]
- [tex]\((-3)^2 = 9\)[/tex], so [tex]\(27(9) = 243\)[/tex]
- [tex]\(9(-3) = -27\)[/tex]
4. Substitute these values back into the expression:
[tex]\[
-189 + 243 - 27 - 27
\][/tex]
5. Add and subtract the values:
- Combine: [tex]\(-189 + 243 = 54\)[/tex]
- Result: [tex]\(54 - 27 - 27 = 0\)[/tex]
The result is [tex]\(0\)[/tex], indicating that substituting [tex]\(-3\)[/tex] into the polynomial yields zero.
Thus, according to the Factor Theorem, [tex]\(x + 3\)[/tex] is indeed a factor of the polynomial [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex].
For the polynomial [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex], we need to check if [tex]\(x + 3\)[/tex] is a factor. Therefore, we substitute [tex]\(x = -3\)[/tex] into the polynomial. Let's evaluate the polynomial at [tex]\(x = -3\)[/tex]:
1. Start with the polynomial: [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex].
2. Substitute [tex]\(x = -3\)[/tex] into the polynomial:
[tex]\[
7(-3)^3 + 27(-3)^2 + 9(-3) - 27
\][/tex]
3. Calculate each term:
- [tex]\((-3)^3 = -27\)[/tex], so [tex]\(7(-27) = -189\)[/tex]
- [tex]\((-3)^2 = 9\)[/tex], so [tex]\(27(9) = 243\)[/tex]
- [tex]\(9(-3) = -27\)[/tex]
4. Substitute these values back into the expression:
[tex]\[
-189 + 243 - 27 - 27
\][/tex]
5. Add and subtract the values:
- Combine: [tex]\(-189 + 243 = 54\)[/tex]
- Result: [tex]\(54 - 27 - 27 = 0\)[/tex]
The result is [tex]\(0\)[/tex], indicating that substituting [tex]\(-3\)[/tex] into the polynomial yields zero.
Thus, according to the Factor Theorem, [tex]\(x + 3\)[/tex] is indeed a factor of the polynomial [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex].
