Answer :
Sure! Let's work through the polynomial division step by step to divide [tex]\(\left( 3x^5 + 15x^4 + 26x^3 + 23x^2 - 4x \right) \)[/tex] by [tex]\( (x + 3) \)[/tex].
### Step-by-Step Solution
1. Write down the division in long division format:
[tex]\( \begin{array}{r}
3x^4 + 6x^3 + 8x^2 - x - 1 \\
-------------------------- \)[/tex] \\
[tex]\( x + 3 \mid 3x^5 + 15x^4 + 26x^3 + 23x^2 - 4x \)[/tex] \\
\end{array} \)
2. Divide the first term of the polynomial [tex]\(3x^5\)[/tex] by the first term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{3x^5}{x} = 3x^4
\][/tex]
Write [tex]\(3x^4\)[/tex] above the division line.
3. Multiply [tex]\(3x^4\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
3x^4 \cdot (x + 3) = 3x^5 + 9x^4
\][/tex]
Write this below the original polynomial and subtract:
[tex]\[
(3x^5 + 15x^4 + 26x^3 + 23x^2 - 4x) - (3x^5 + 9x^4) = 6x^4 + 26x^3 + 23x^2 - 4x
\][/tex]
4. Next, divide [tex]\(6x^4\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{6x^4}{x} = 6x^3
\][/tex]
Write [tex]\(6x^3\)[/tex] above the division line next to the [tex]\(3x^4\)[/tex].
5. Multiply [tex]\(6x^3\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
6x^3 \cdot (x + 3) = 6x^4 + 18x^3
\][/tex]
Write this below the current polynomial and subtract:
[tex]\[
(6x^4 + 26x^3 + 23x^2 - 4x) - (6x^4 + 18x^3) = 8x^3 + 23x^2 - 4x
\][/tex]
6. Next, divide [tex]\(8x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{8x^3}{x} = 8x^2
\][/tex]
Write [tex]\(8x^2\)[/tex] above the division line next to the [tex]\(6x^3\)[/tex].
7. Multiply [tex]\(8x^2\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
8x^2 \cdot (x + 3) = 8x^3 + 24x^2
\][/tex]
Write this below the current polynomial and subtract:
[tex]\[
(8x^3 + 23x^2 - 4x) - (8x^3 + 24x^2) = -x^2 - 4x
\][/tex]
8. Next, divide [tex]\(-x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{-x^2}{x} = -x
\][/tex]
Write [tex]\(-x\)[/tex] above the division line next to [tex]\(8x^2\)[/tex].
9. Multiply [tex]\(-x\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
-x \cdot (x + 3) = -x^2 - 3x
\][/tex]
Write this below the current polynomial and subtract:
[tex]\[
(-x^2 - 4x) - (-x^2 - 3x) = -x
\][/tex]
10. Next, divide [tex]\(-x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{-x}{x} = -1
\][/tex]
Write [tex]\(-1\)[/tex] above the division line next to [tex]\(-x\)[/tex].
11. Multiply [tex]\(-1\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
-1 \cdot (x + 3) = -x - 3
\][/tex]
Write this below the current polynomial and subtract:
[tex]\[
(-x) - (-x - 3) = 3
\][/tex]
### Final Result:
The quotient is [tex]\(3x^4 + 6x^3 + 8x^2 - x - 1\)[/tex] and the remainder is [tex]\(3\)[/tex].
Thus, the result of the division [tex]\(\left( 3x^5 + 15x^4 + 26x^3 + 23x^2 - 4x \right) \div (x + 3) \)[/tex] is:
[tex]\[
\boxed{3x^4 + 6x^3 + 8x^2 - x - 1 \text{ with a remainder of } 3}
\][/tex]
### Step-by-Step Solution
1. Write down the division in long division format:
[tex]\( \begin{array}{r}
3x^4 + 6x^3 + 8x^2 - x - 1 \\
-------------------------- \)[/tex] \\
[tex]\( x + 3 \mid 3x^5 + 15x^4 + 26x^3 + 23x^2 - 4x \)[/tex] \\
\end{array} \)
2. Divide the first term of the polynomial [tex]\(3x^5\)[/tex] by the first term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{3x^5}{x} = 3x^4
\][/tex]
Write [tex]\(3x^4\)[/tex] above the division line.
3. Multiply [tex]\(3x^4\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
3x^4 \cdot (x + 3) = 3x^5 + 9x^4
\][/tex]
Write this below the original polynomial and subtract:
[tex]\[
(3x^5 + 15x^4 + 26x^3 + 23x^2 - 4x) - (3x^5 + 9x^4) = 6x^4 + 26x^3 + 23x^2 - 4x
\][/tex]
4. Next, divide [tex]\(6x^4\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{6x^4}{x} = 6x^3
\][/tex]
Write [tex]\(6x^3\)[/tex] above the division line next to the [tex]\(3x^4\)[/tex].
5. Multiply [tex]\(6x^3\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
6x^3 \cdot (x + 3) = 6x^4 + 18x^3
\][/tex]
Write this below the current polynomial and subtract:
[tex]\[
(6x^4 + 26x^3 + 23x^2 - 4x) - (6x^4 + 18x^3) = 8x^3 + 23x^2 - 4x
\][/tex]
6. Next, divide [tex]\(8x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{8x^3}{x} = 8x^2
\][/tex]
Write [tex]\(8x^2\)[/tex] above the division line next to the [tex]\(6x^3\)[/tex].
7. Multiply [tex]\(8x^2\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
8x^2 \cdot (x + 3) = 8x^3 + 24x^2
\][/tex]
Write this below the current polynomial and subtract:
[tex]\[
(8x^3 + 23x^2 - 4x) - (8x^3 + 24x^2) = -x^2 - 4x
\][/tex]
8. Next, divide [tex]\(-x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{-x^2}{x} = -x
\][/tex]
Write [tex]\(-x\)[/tex] above the division line next to [tex]\(8x^2\)[/tex].
9. Multiply [tex]\(-x\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
-x \cdot (x + 3) = -x^2 - 3x
\][/tex]
Write this below the current polynomial and subtract:
[tex]\[
(-x^2 - 4x) - (-x^2 - 3x) = -x
\][/tex]
10. Next, divide [tex]\(-x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{-x}{x} = -1
\][/tex]
Write [tex]\(-1\)[/tex] above the division line next to [tex]\(-x\)[/tex].
11. Multiply [tex]\(-1\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[
-1 \cdot (x + 3) = -x - 3
\][/tex]
Write this below the current polynomial and subtract:
[tex]\[
(-x) - (-x - 3) = 3
\][/tex]
### Final Result:
The quotient is [tex]\(3x^4 + 6x^3 + 8x^2 - x - 1\)[/tex] and the remainder is [tex]\(3\)[/tex].
Thus, the result of the division [tex]\(\left( 3x^5 + 15x^4 + 26x^3 + 23x^2 - 4x \right) \div (x + 3) \)[/tex] is:
[tex]\[
\boxed{3x^4 + 6x^3 + 8x^2 - x - 1 \text{ with a remainder of } 3}
\][/tex]
