Answer :
To find the remainder when the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex], we need to perform polynomial division. Here's how it works step by step:
1. Division Setup: We're dividing a cubic polynomial by a quadratic polynomial. This means that our quotient will be a linear polynomial, and the remainder will be of lower degree than the divisor (which is [tex]\(x^2 + 3x + 3\)[/tex]).
2. Divide the Leading Terms: First, divide the leading term of the dividend, [tex]\(3x^3\)[/tex], by the leading term of the divisor, [tex]\(x^2\)[/tex]. This gives us:
[tex]\[
\frac{3x^3}{x^2} = 3x
\][/tex]
This becomes the first term of the quotient.
3. Multiply and Subtract: Multiply the whole divisor [tex]\(x^2 + 3x + 3\)[/tex] by the result from step 2 (which is [tex]\(3x\)[/tex]):
[tex]\[
3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Process: Now divide the new leading term, [tex]\(-11x^2\)[/tex], by the leading term of the divisor, [tex]\(x^2\)[/tex]:
[tex]\[
\frac{-11x^2}{x^2} = -11
\][/tex]
This becomes the next term of the quotient.
5. Multiply and Subtract Again: Multiply the entire divisor by [tex]\(-11\)[/tex]:
[tex]\[
-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
Subtract this from the remaining polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Identify the Remainder: Since the result, [tex]\(28x + 30\)[/tex], is of lower degree than the divisor [tex]\(x^2 + 3x + 3\)[/tex], this is the remainder of the division.
Therefore, the remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex] is [tex]\(28x + 30\)[/tex].
1. Division Setup: We're dividing a cubic polynomial by a quadratic polynomial. This means that our quotient will be a linear polynomial, and the remainder will be of lower degree than the divisor (which is [tex]\(x^2 + 3x + 3\)[/tex]).
2. Divide the Leading Terms: First, divide the leading term of the dividend, [tex]\(3x^3\)[/tex], by the leading term of the divisor, [tex]\(x^2\)[/tex]. This gives us:
[tex]\[
\frac{3x^3}{x^2} = 3x
\][/tex]
This becomes the first term of the quotient.
3. Multiply and Subtract: Multiply the whole divisor [tex]\(x^2 + 3x + 3\)[/tex] by the result from step 2 (which is [tex]\(3x\)[/tex]):
[tex]\[
3x \cdot (x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x) = -11x^2 - 5x - 3
\][/tex]
4. Repeat the Process: Now divide the new leading term, [tex]\(-11x^2\)[/tex], by the leading term of the divisor, [tex]\(x^2\)[/tex]:
[tex]\[
\frac{-11x^2}{x^2} = -11
\][/tex]
This becomes the next term of the quotient.
5. Multiply and Subtract Again: Multiply the entire divisor by [tex]\(-11\)[/tex]:
[tex]\[
-11 \cdot (x^2 + 3x + 3) = -11x^2 - 33x - 33
\][/tex]
Subtract this from the remaining polynomial:
[tex]\[
(-11x^2 - 5x - 3) - (-11x^2 - 33x - 33) = 28x + 30
\][/tex]
6. Identify the Remainder: Since the result, [tex]\(28x + 30\)[/tex], is of lower degree than the divisor [tex]\(x^2 + 3x + 3\)[/tex], this is the remainder of the division.
Therefore, the remainder when [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by [tex]\(x^2 + 3x + 3\)[/tex] is [tex]\(28x + 30\)[/tex].
