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. The Factor Theorem states that [tex]\( x-a \)[/tex] is a factor of a polynomial if and only if substituting [tex]\( a \)[/tex] into the polynomial yields zero.
For the polynomial [tex]\( 7x^3 + 27x^2 + 9x - 27 \)[/tex], we want to check if [tex]\( x+3 \)[/tex] is a factor. Therefore, we substitute [tex]\(-3\)[/tex] (the opposite of [tex]\( +3 \)[/tex]) into the polynomial and see if the result is zero.
Let's calculate [tex]\( f(-3) \)[/tex]:
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 step-by-step:
- [tex]\( 7(-3)^3 = 7(-27) = -189 \)[/tex]
- [tex]\( 27(-3)^2 = 27(9) = 243 \)[/tex]
- [tex]\( 9(-3) = -27 \)[/tex]
- The last term is [tex]\(-27\)[/tex].
3. Now, substitute these values back into the equation:
[tex]\[
f(-3) = -189 + 243 - 27 - 27
\][/tex]
4. Simplify:
[tex]\[
f(-3) = -189 + 243 - 27 - 27 = 0
\][/tex]
Since [tex]\( f(-3) = 0 \)[/tex], the remainder is zero when you substitute [tex]\(-3\)[/tex] into the polynomial. This means [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 want to check if [tex]\( x+3 \)[/tex] is a factor. Therefore, we substitute [tex]\(-3\)[/tex] (the opposite of [tex]\( +3 \)[/tex]) into the polynomial and see if the result is zero.
Let's calculate [tex]\( f(-3) \)[/tex]:
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 step-by-step:
- [tex]\( 7(-3)^3 = 7(-27) = -189 \)[/tex]
- [tex]\( 27(-3)^2 = 27(9) = 243 \)[/tex]
- [tex]\( 9(-3) = -27 \)[/tex]
- The last term is [tex]\(-27\)[/tex].
3. Now, substitute these values back into the equation:
[tex]\[
f(-3) = -189 + 243 - 27 - 27
\][/tex]
4. Simplify:
[tex]\[
f(-3) = -189 + 243 - 27 - 27 = 0
\][/tex]
Since [tex]\( f(-3) = 0 \)[/tex], the remainder is zero when you substitute [tex]\(-3\)[/tex] into the polynomial. This means [tex]\( x+3 \)[/tex] is indeed a factor of the polynomial [tex]\( 7x^3 + 27x^2 + 9x - 27 \)[/tex].
