To find the remainder of the division of [tex]\(\frac{5x^3 + 7x + 5}{x + 2}\)[/tex], we need to use polynomial long division or synthetic division. Here's the detailed step-by-step process for polynomial long division:
1. Set up the division: We write [tex]\(5x^3 + 7x + 5\)[/tex] as the dividend and [tex]\(x + 2\)[/tex] as the divisor.
[tex]\[
\text{Dividend: } (5x^3 + 0x^2 + 7x + 5) \\
\text{Divisor: } (x + 2)
\][/tex]
2. Divide the leading term of the dividend by the leading term of the divisor: Divide [tex]\(5x^3\)[/tex] by [tex]\(x\)[/tex].
[tex]\[
\frac{5x^3}{x} = 5x^2
\][/tex]
3. Multiply the entire divisor by the result from step 2: Multiply [tex]\(x + 2\)[/tex] by [tex]\(5x^2\)[/tex].
[tex]\[
(5x^2)(x + 2) = 5x^3 + 10x^2
\][/tex]
4. Subtract this result from the original dividend:
[tex]\[
(5x^3 + 0x^2 + 7x + 5) - (5x^3 + 10x^2) = 0x^3 - 10x^2 + 7x + 5 = -10x^2 + 7x + 5
\][/tex]
5. Repeat the process with the new polynomial:
- Divide [tex]\(-10x^2\)[/tex] by [tex]\(x\)[/tex].
[tex]\[
\frac{-10x^2}{x} = -10x
\][/tex]
- Multiply the entire divisor by the result.
[tex]\[
(-10x)(x + 2) = -10x^2 - 20x
\][/tex]
- Subtract this from [tex]\(-10x^2 + 7x + 5\)[/tex].
[tex]\[
(-10x^2 + 7x + 5) - (-10x^2 - 20x) = 0x^2 + 27x + 5 = 27x + 5
\][/tex]
6. Continue the process:
- Divide [tex]\(27x\)[/tex] by [tex]\(x\)[/tex].
[tex]\[
\frac{27x}{x} = 27
\][/tex]
- Multiply the entire divisor by the result.
[tex]\[
(27)(x + 2) = 27x + 54
\][/tex]
- Subtract this from [tex]\(27x + 5\)[/tex].
[tex]\[
(27x + 5) - (27x + 54) = 5 - 54 = -49
\][/tex]
Since [tex]\(x + 2\)[/tex] is now a degree higher than any remaining polynomial term, the remainder is the last result obtained:
Remainder: [tex]\(-49\)[/tex]
Therefore, the remainder when dividing [tex]\(5x^3 + 7x + 5\)[/tex] by [tex]\(x + 2\)[/tex] is [tex]\(-49\)[/tex].
