Answer :
To solve the problem [tex]\(\frac{3x^4 + 19x^3 + 30x^2 + 9x + 27}{x+3}\)[/tex], we can perform polynomial long division. Here's how it works step-by-step:
1. Divide the first terms:
Divide the leading term of the dividend [tex]\(3x^4\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{3x^4}{x} = 3x^3
\][/tex]
2. Multiply and subtract:
Multiply the entire divisor [tex]\(x + 3\)[/tex] by the result from step 1, which is [tex]\(3x^3\)[/tex], and subtract from the original polynomial:
[tex]\[
(3x^4 + 19x^3 + 30x^2 + 9x + 27) - (3x^3 \cdot (x + 3)) = (3x^4 + 19x^3 + 30x^2 + 9x + 27) - (3x^4 + 9x^3)
\][/tex]
This simplifies to:
[tex]\[
10x^3 + 30x^2 + 9x + 27
\][/tex]
3. Repeat the process:
Divide the new leading term [tex]\(10x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{10x^3}{x} = 10x^2
\][/tex]
Multiply and subtract again:
[tex]\[
(10x^3 + 30x^2 + 9x + 27) - (10x^2 \cdot (x + 3)) = (10x^3 + 30x^2 + 9x + 27) - (10x^3 + 30x^2)
\][/tex]
This results in:
[tex]\[
9x + 27
\][/tex]
4. Continue dividing:
Divide the next leading term [tex]\(9x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{9x}{x} = 9
\][/tex]
Apply the same steps:
[tex]\[
(9x + 27) - (9 \cdot (x + 3)) = (9x + 27) - (9x + 27)
\][/tex]
This results in a remainder of 0.
5. Bring it all together:
The quotient is [tex]\(3x^3 + 10x^2 + 9\)[/tex] and the remainder is 0. Therefore, the expression simplifies to:
[tex]\[
3x^3 + 10x^2 + 9
\][/tex]
Thus, the solution to [tex]\(\frac{3x^4 + 19x^3 + 30x^2 + 9x + 27}{x+3}\)[/tex] is [tex]\(3x^3 + 10x^2 + 9\)[/tex].
1. Divide the first terms:
Divide the leading term of the dividend [tex]\(3x^4\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{3x^4}{x} = 3x^3
\][/tex]
2. Multiply and subtract:
Multiply the entire divisor [tex]\(x + 3\)[/tex] by the result from step 1, which is [tex]\(3x^3\)[/tex], and subtract from the original polynomial:
[tex]\[
(3x^4 + 19x^3 + 30x^2 + 9x + 27) - (3x^3 \cdot (x + 3)) = (3x^4 + 19x^3 + 30x^2 + 9x + 27) - (3x^4 + 9x^3)
\][/tex]
This simplifies to:
[tex]\[
10x^3 + 30x^2 + 9x + 27
\][/tex]
3. Repeat the process:
Divide the new leading term [tex]\(10x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{10x^3}{x} = 10x^2
\][/tex]
Multiply and subtract again:
[tex]\[
(10x^3 + 30x^2 + 9x + 27) - (10x^2 \cdot (x + 3)) = (10x^3 + 30x^2 + 9x + 27) - (10x^3 + 30x^2)
\][/tex]
This results in:
[tex]\[
9x + 27
\][/tex]
4. Continue dividing:
Divide the next leading term [tex]\(9x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{9x}{x} = 9
\][/tex]
Apply the same steps:
[tex]\[
(9x + 27) - (9 \cdot (x + 3)) = (9x + 27) - (9x + 27)
\][/tex]
This results in a remainder of 0.
5. Bring it all together:
The quotient is [tex]\(3x^3 + 10x^2 + 9\)[/tex] and the remainder is 0. Therefore, the expression simplifies to:
[tex]\[
3x^3 + 10x^2 + 9
\][/tex]
Thus, the solution to [tex]\(\frac{3x^4 + 19x^3 + 30x^2 + 9x + 27}{x+3}\)[/tex] is [tex]\(3x^3 + 10x^2 + 9\)[/tex].
