To find the remainder when [tex]\( x^2 + 5 \)[/tex] is divided by [tex]\( x + 1 \)[/tex], we can use the process of polynomial long division.
1. Set up the division:
- Dividend (the polynomial to be divided): [tex]\( x^2 + 5 \)[/tex]
- Divisor (the polynomial we are dividing by): [tex]\( x + 1 \)[/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
So, the first term in the quotient is [tex]\( x \)[/tex].
3. Multiply the entire divisor [tex]\( x + 1 \)[/tex] by this term [tex]\( x \)[/tex]:
[tex]\[
x \cdot (x + 1) = x^2 + x
\][/tex]
4. Subtract this result from the original dividend:
[tex]\[
(x^2 + 5) - (x^2 + x) = (x^2 + 5) - x^2 - x = 5 - x
\][/tex]
After this subtraction, the new polynomial is [tex]\( -x + 5 \)[/tex].
5. Divide the leading term of the new polynomial by the leading term of the divisor:
[tex]\[
\frac{-x}{x} = -1
\][/tex]
So, the next term in the quotient is [tex]\( -1 \)[/tex].
6. Multiply the entire divisor [tex]\( x + 1 \)[/tex] by this term [tex]\( -1 \)[/tex]:
[tex]\[
-1 \cdot (x + 1) = -x - 1
\][/tex]
7. Subtract this result from the new polynomial:
[tex]\[
(-x + 5) - (-x - 1) = (-x + 5) + x + 1 = 6
\][/tex]
After performing these steps, we end up with a quotient of [tex]\( x - 1 \)[/tex] and a remainder of [tex]\( 6 \)[/tex].
Therefore, the remainder when [tex]\( x^2 + 5 \)[/tex] is divided by [tex]\( x + 1 \)[/tex] is [tex]\( 6 \)[/tex].
