To find the multiple of [tex]\(x^3\)[/tex] that must be added to the expression [tex]\(3x^5 + 3x^4 + 8x^2 + 48\)[/tex] so that the result has a factor of [tex]\((x + 2)\)[/tex], we need to ensure that the polynomial is divisible by [tex]\((x + 2)\)[/tex]. This implies that the polynomial must yield a zero remainder when divided by [tex]\((x + 2)\)[/tex].
Here's a step-by-step solution:
1. Substitute [tex]\(x = -2\)[/tex] into the polynomial:
We substitute [tex]\(x = -2\)[/tex] into the given polynomial expression [tex]\(3x^5 + 3x^4 + 8x^2 + 48\)[/tex] to find out if it equals zero or if there is a remainder.
[tex]\[
3(-2)^5 + 3(-2)^4 + 8(-2)^2 + 48
\][/tex]
2. Calculate each term individually:
[tex]\[
3(-32) + 3(16) + 8(4) + 48
\][/tex]
[tex]\[
-96 + 48 + 32 + 48
\][/tex]
3. Sum the calculated values:
[tex]\[
-96 + 48 = -48
\][/tex]
[tex]\[
-48 + 32 = -16
\][/tex]
[tex]\[
-16 + 48 = 32
\][/tex]
Thus, when [tex]\(x = -2\)[/tex], the polynomial evaluates to [tex]\(32\)[/tex]. This remainder needs to be canceled out to make the polynomial divisible by [tex]\((x + 2)\)[/tex].
4. Determine the missing term:
We need to add a term to the polynomial that will eliminate this remainder. Since the polynomial needs a term involving [tex]\(x^3\)[/tex], let's call this term [tex]\(C x^3\)[/tex].
5. Formulate the modified polynomial:
The modified polynomial will be:
[tex]\[
3x^5 + 3x^4 + C x^3 + 8x^2 + 48
\][/tex]
6. Substitute [tex]\(x = -2\)[/tex] into the modified polynomial:
To find [tex]\(C\)[/tex], we substitute [tex]\(x = -2\)[/tex] into the polynomial including the new [tex]\(C x^3\)[/tex] term and set the result equal to zero (since it should now be divisible by [tex]\((x + 2)\)[/tex]):
[tex]\[
3(-2)^5 + 3(-2)^4 + C(-2)^3 + 8(-2)^2 + 48 = 0
\][/tex]
[tex]\[
-96 + 48 + C(-8) + 32 + 48 = 0
\][/tex]
[tex]\[
32 + C(-8) = 0
\][/tex]
7. Solve for [tex]\(C\)[/tex]:
[tex]\[
32 - 8C = 0
\][/tex]
[tex]\[
-8C = -32
\][/tex]
[tex]\[
C = 4
\][/tex]
Therefore, the multiple of [tex]\(x^3\)[/tex] that must be added to the polynomial [tex]\(3 x^5 + 3 x^4 + 8 x^2 + 48\)[/tex] so that it is divisible by [tex]\((x + 2)\)[/tex] is [tex]\(-32\)[/tex].
