Answer :
To solve this problem, we need to find the remainder when the polynomial [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] is divided by the polynomial [tex]\(x^2 + 3x + 3\)[/tex].
Here's a step-by-step explanation:
1. Understand Polynomial Division: When dividing polynomials, the goal is to express the dividend as a combination of the divisor and a remainder that has a degree less than the divisor. For this problem, the dividend is [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] and the divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Perform Polynomial Long Division:
- Step 1: Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{3x^3}{x^2} = 3x
\][/tex]
- Step 2: Multiply the entire divisor by [tex]\(3x\)[/tex] and subtract from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x(x^2 + 3x + 3)) = (3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x)
\][/tex]
- Step 3: Simplify the result:
[tex]\[
-2x^2 + 4x - 3 - 9x^2 - 9x = -11x^2 - 5x - 3
\][/tex]
- Step 4: Divide the new leading term by the leading term of the divisor:
[tex]\[
\frac{-11x^2}{x^2} = -11
\][/tex]
- Step 5: Multiply the entire divisor by [tex]\(-11\)[/tex] and subtract:
[tex]\[
(-11x^2 - 5x - 3) - (-11)(x^2 + 3x + 3) = -11x^2 - 5x - 3 + 11x^2 + 33x + 33
\][/tex]
- Step 6: Simplify the result to get the remainder:
[tex]\[
28x + 30
\][/tex]
3. Conclusion: 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].
So, the answer is [tex]\(28x + 30\)[/tex].
Here's a step-by-step explanation:
1. Understand Polynomial Division: When dividing polynomials, the goal is to express the dividend as a combination of the divisor and a remainder that has a degree less than the divisor. For this problem, the dividend is [tex]\(3x^3 - 2x^2 + 4x - 3\)[/tex] and the divisor is [tex]\(x^2 + 3x + 3\)[/tex].
2. Perform Polynomial Long Division:
- Step 1: Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{3x^3}{x^2} = 3x
\][/tex]
- Step 2: Multiply the entire divisor by [tex]\(3x\)[/tex] and subtract from the dividend:
[tex]\[
(3x^3 - 2x^2 + 4x - 3) - (3x(x^2 + 3x + 3)) = (3x^3 - 2x^2 + 4x - 3) - (3x^3 + 9x^2 + 9x)
\][/tex]
- Step 3: Simplify the result:
[tex]\[
-2x^2 + 4x - 3 - 9x^2 - 9x = -11x^2 - 5x - 3
\][/tex]
- Step 4: Divide the new leading term by the leading term of the divisor:
[tex]\[
\frac{-11x^2}{x^2} = -11
\][/tex]
- Step 5: Multiply the entire divisor by [tex]\(-11\)[/tex] and subtract:
[tex]\[
(-11x^2 - 5x - 3) - (-11)(x^2 + 3x + 3) = -11x^2 - 5x - 3 + 11x^2 + 33x + 33
\][/tex]
- Step 6: Simplify the result to get the remainder:
[tex]\[
28x + 30
\][/tex]
3. Conclusion: 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].
So, the answer is [tex]\(28x + 30\)[/tex].
