To find the remainder when dividing [tex]\( 5x^3 + 7x + 5 \)[/tex] by [tex]\( x + 2 \)[/tex], we will perform polynomial division, a method similar to long division for numbers. Here is a step-by-step solution:
### Step-by-Step Solution
1. Set Up the Polynomial Division:
The dividend (the polynomial we are dividing) is [tex]\( 5x^3 + 7x + 5 \)[/tex].
The divisor (the polynomial we are dividing by) is [tex]\( x + 2 \)[/tex].
2. Perform the Division:
Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[
\frac{5x^3}{x} = 5x^2
\][/tex]
So, the first term of the quotient is [tex]\( 5x^2 \)[/tex].
Step 2: Multiply the entire divisor [tex]\( x + 2 \)[/tex] by this term [tex]\( 5x^2 \)[/tex]:
[tex]\[
(x + 2) \cdot 5x^2 = 5x^3 + 10x^2
\][/tex]
Step 3: Subtract this result from the original dividend:
[tex]\[
(5x^3 + 0x^2 + 7x + 5) - (5x^3 + 10x^2) = -10x^2 + 7x + 5
\][/tex]
Step 4: Repeat the process with the new polynomial [tex]\(-10x^2 + 7x + 5\)[/tex].
Divide the leading term of the new dividend by the leading term of the divisor:
[tex]\[
\frac{-10x^2}{x} = -10x
\][/tex]
So, the next term of the quotient is [tex]\(-10x\)[/tex].
Step 5: Multiply the divisor by this new term [tex]\(-10x\)[/tex]:
[tex]\[
(x + 2) \cdot (-10x) = -10x^2 - 20x
\][/tex]
Step 6: Subtract this result from the new dividend:
[tex]\[
(-10x^2 + 7x + 5) - (-10x^2 - 20x) = 27x + 5
\][/tex]
Step 7: Repeat the process with the new polynomial [tex]\(27x + 5\)[/tex].
Divide the leading term of the new dividend by the leading term of the divisor:
[tex]\[
\frac{27x}{x} = 27
\][/tex]
So, the next term of the quotient is [tex]\(27\)[/tex].
Step 8: Multiply the divisor by this new term [tex]\(27\)[/tex]:
[tex]\[
(x + 2) \cdot 27 = 27x + 54
\][/tex]
Step 9: Subtract this result from the new dividend:
[tex]\[
(27x + 5) - (27x + 54) = 5 - 54 = -49
\][/tex]
3. Result:
At this point, no further division is possible because the degree of the new polynomial is less than the degree of the divisor [tex]\( x + 2 \)[/tex]. Thus, the remainder is:
[tex]\[
-49
\][/tex]
### Final Answer
The remainder of the division of [tex]\( \frac{5x^3 + 7x + 5}{x + 2} \)[/tex] is [tex]\( -49 \)[/tex].
