To find the remainder of the polynomial division [tex]\(\frac{5x^3 + 7x + 5}{x + 2}\)[/tex], we need to perform polynomial long division. We will divide the polynomial [tex]\(5x^3 + 7x + 5\)[/tex] by the divisor [tex]\(x + 2\)[/tex].
1. Setup the Division:
We arrange the terms in descending order of power.
[tex]\[
\frac{5x^3 + 0x^2 + 7x + 5}{x + 2}
\][/tex]
2. Divide the Leading Terms:
We begin by dividing the leading term of the numerator [tex]\(5x^3\)[/tex] by the leading term of the denominator [tex]\(x\)[/tex]:
[tex]\[
\frac{5x^3}{x} = 5x^2
\][/tex]
This is the first term of our quotient.
3. Multiply and Subtract:
Multiply [tex]\(5x^2\)[/tex] by [tex]\(x + 2\)[/tex]:
[tex]\[
5x^2 \cdot (x + 2) = 5x^3 + 10x^2
\][/tex]
Subtract this result from the original polynomial:
[tex]\[
(5x^3 + 0x^2 + 7x + 5) - (5x^3 + 10x^2) = -10x^2 + 7x + 5
\][/tex]
4. Repeat the Process:
Next, divide the leading term of the new polynomial [tex]\(-10x^2\)[/tex] by the leading term of the denominator [tex]\(x\)[/tex]:
[tex]\[
\frac{-10x^2}{x} = -10x
\][/tex]
This is the second term of our quotient.
Multiply [tex]\(-10x\)[/tex] by [tex]\(x + 2\)[/tex]:
[tex]\[
-10x \cdot (x + 2) = -10x^2 - 20x
\][/tex]
Subtract this result from the new polynomial:
[tex]\[
(-10x^2 + 7x + 5) - (-10x^2 - 20x) = 27x + 5
\][/tex]
5. Repeat Again:
Now, divide the leading term of the new polynomial [tex]\(27x\)[/tex] by the leading term of the denominator [tex]\(x\)[/tex]:
[tex]\[
\frac{27x}{x} = 27
\][/tex]
This is the third term of our quotient.
Multiply [tex]\(27\)[/tex] by [tex]\(x + 2\)[/tex]:
[tex]\[
27 \cdot (x + 2) = 27x + 54
\][/tex]
Subtract this result from the new polynomial:
[tex]\[
(27x + 5) - (27x + 54) = -49
\][/tex]
6. Final Quotient and Remainder:
The quotient of the division is [tex]\(5x^2 - 10x + 27\)[/tex] and the remainder is [tex]\(-49\)[/tex].
So the remainder when dividing [tex]\(5x^3 + 7x + 5\)[/tex] by [tex]\(x + 2\)[/tex] is [tex]\(\boxed{-49}\)[/tex].
