Answer :
To determine whether [tex]\(x + 3\)[/tex] is a factor of the polynomial [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex], we can use the Remainder Theorem. The theorem states that a polynomial [tex]\(f(x)\)[/tex] has a factor [tex]\(x - c\)[/tex] if and only if [tex]\(f(c) = 0\)[/tex].
In our case, we want to check if [tex]\(x + 3\)[/tex] is a factor, which means we need to evaluate the polynomial at [tex]\(x = -3\)[/tex]. If the result is 0, then [tex]\(x + 3\)[/tex] is a factor.
1. Substitute [tex]\(x = -3\)[/tex] into the polynomial:
[tex]\[
f(-3) = 7(-3)^3 + 27(-3)^2 + 9(-3) - 27
\][/tex]
2. Calculate each term:
- [tex]\((-3)^3 = -27\)[/tex], so [tex]\(7(-3)^3 = 7 \times -27 = -189\)[/tex]
- [tex]\((-3)^2 = 9\)[/tex], so [tex]\(27(-3)^2 = 27 \times 9 = 243\)[/tex]
- [tex]\(9(-3) = -27\)[/tex]
- The constant term is [tex]\(-27\)[/tex]
3. Combine these results:
[tex]\[
f(-3) = -189 + 243 - 27 - 27
\][/tex]
4. Simplify the expression:
[tex]\[
f(-3) = 54 - 54 = 0
\][/tex]
Since the value of [tex]\(f(-3)\)[/tex] is 0, it means that [tex]\(x + 3\)[/tex] is indeed a factor of the polynomial [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex].
Therefore, the answer is: Yes, [tex]\(x + 3\)[/tex] is a factor of the polynomial.
In our case, we want to check if [tex]\(x + 3\)[/tex] is a factor, which means we need to evaluate the polynomial at [tex]\(x = -3\)[/tex]. If the result is 0, then [tex]\(x + 3\)[/tex] is a factor.
1. Substitute [tex]\(x = -3\)[/tex] into the polynomial:
[tex]\[
f(-3) = 7(-3)^3 + 27(-3)^2 + 9(-3) - 27
\][/tex]
2. Calculate each term:
- [tex]\((-3)^3 = -27\)[/tex], so [tex]\(7(-3)^3 = 7 \times -27 = -189\)[/tex]
- [tex]\((-3)^2 = 9\)[/tex], so [tex]\(27(-3)^2 = 27 \times 9 = 243\)[/tex]
- [tex]\(9(-3) = -27\)[/tex]
- The constant term is [tex]\(-27\)[/tex]
3. Combine these results:
[tex]\[
f(-3) = -189 + 243 - 27 - 27
\][/tex]
4. Simplify the expression:
[tex]\[
f(-3) = 54 - 54 = 0
\][/tex]
Since the value of [tex]\(f(-3)\)[/tex] is 0, it means that [tex]\(x + 3\)[/tex] is indeed a factor of the polynomial [tex]\(7x^3 + 27x^2 + 9x - 27\)[/tex].
Therefore, the answer is: Yes, [tex]\(x + 3\)[/tex] is a factor of the polynomial.
