Answer :
Sure! Let's perform the polynomial division step-by-step to find the quotient and remainder of [tex]\((3x^5 - 23x^4 + 15x^3 + 40x^2 - 4x - 43) \div (3x - 5)\)[/tex].
Step 1: Set Up the Division
We are dividing [tex]\(3x^5 - 23x^4 + 15x^3 + 40x^2 - 4x - 43\)[/tex] by [tex]\(3x - 5\)[/tex].
Step 2: Divide the First Terms
- Divide the leading term of the dividend, [tex]\(3x^5\)[/tex], by the leading term of the divisor, [tex]\(3x\)[/tex]:
[tex]\[
\frac{3x^5}{3x} = x^4
\][/tex]
Step 3: Multiply and Subtract
- Multiply the entire divisor [tex]\(3x - 5\)[/tex] by [tex]\(x^4\)[/tex]:
[tex]\[
(3x - 5) \cdot x^4 = 3x^5 - 5x^4
\][/tex]
- Subtract this from the original polynomial:
[tex]\[
(3x^5 - 23x^4 + 15x^3 + 40x^2 - 4x - 43) - (3x^5 - 5x^4) = -18x^4 + 15x^3 + 40x^2 - 4x - 43
\][/tex]
Step 4: Repeat the Process
1. Divide the next leading term:
[tex]\[
\frac{-18x^4}{3x} = -6x^3
\][/tex]
2. Multiply and subtract:
[tex]\[
(3x - 5) \cdot (-6x^3) = -18x^4 + 30x^3
\][/tex]
[tex]\[
(-18x^4 + 15x^3 + 40x^2 - 4x - 43) - (-18x^4 + 30x^3) = -15x^3 + 40x^2 - 4x - 43
\][/tex]
3. Continue this pattern:
Divide the next leading term:
[tex]\[
\frac{-15x^3}{3x} = -5x^2
\][/tex]
Multiply and subtract:
[tex]\[
(3x - 5) \cdot (-5x^2) = -15x^3 + 25x^2
\][/tex]
[tex]\[
(-15x^3 + 40x^2 - 4x - 43) - (-15x^3 + 25x^2) = 15x^2 - 4x - 43
\][/tex]
Divide the next leading term:
[tex]\[
\frac{15x^2}{3x} = 5x
\][/tex]
Multiply and subtract:
[tex]\[
(3x - 5) \cdot 5x = 15x^2 - 25x
\][/tex]
[tex]\[
(15x^2 - 4x - 43) - (15x^2 - 25x) = 21x - 43
\][/tex]
Divide the next leading term:
[tex]\[
\frac{21x}{3x} = 7
\][/tex]
Multiply and subtract:
[tex]\[
(3x - 5) \cdot 7 = 21x - 35
\][/tex]
[tex]\[
(21x - 43) - (21x - 35) = -8
\][/tex]
Final Result
The quotient is [tex]\(x^4 - 6x^3 - 5x^2 + 5x + 7\)[/tex] and the remainder is [tex]\(-8\)[/tex].
So, the division of [tex]\(3x^5 - 23x^4 + 15x^3 + 40x^2 - 4x - 43\)[/tex] by [tex]\(3x - 5\)[/tex] results in:
[tex]\[
\text{Quotient: } x^4 - 6x^3 - 5x^2 + 5x + 7, \quad \text{Remainder: } -8
\][/tex]
Step 1: Set Up the Division
We are dividing [tex]\(3x^5 - 23x^4 + 15x^3 + 40x^2 - 4x - 43\)[/tex] by [tex]\(3x - 5\)[/tex].
Step 2: Divide the First Terms
- Divide the leading term of the dividend, [tex]\(3x^5\)[/tex], by the leading term of the divisor, [tex]\(3x\)[/tex]:
[tex]\[
\frac{3x^5}{3x} = x^4
\][/tex]
Step 3: Multiply and Subtract
- Multiply the entire divisor [tex]\(3x - 5\)[/tex] by [tex]\(x^4\)[/tex]:
[tex]\[
(3x - 5) \cdot x^4 = 3x^5 - 5x^4
\][/tex]
- Subtract this from the original polynomial:
[tex]\[
(3x^5 - 23x^4 + 15x^3 + 40x^2 - 4x - 43) - (3x^5 - 5x^4) = -18x^4 + 15x^3 + 40x^2 - 4x - 43
\][/tex]
Step 4: Repeat the Process
1. Divide the next leading term:
[tex]\[
\frac{-18x^4}{3x} = -6x^3
\][/tex]
2. Multiply and subtract:
[tex]\[
(3x - 5) \cdot (-6x^3) = -18x^4 + 30x^3
\][/tex]
[tex]\[
(-18x^4 + 15x^3 + 40x^2 - 4x - 43) - (-18x^4 + 30x^3) = -15x^3 + 40x^2 - 4x - 43
\][/tex]
3. Continue this pattern:
Divide the next leading term:
[tex]\[
\frac{-15x^3}{3x} = -5x^2
\][/tex]
Multiply and subtract:
[tex]\[
(3x - 5) \cdot (-5x^2) = -15x^3 + 25x^2
\][/tex]
[tex]\[
(-15x^3 + 40x^2 - 4x - 43) - (-15x^3 + 25x^2) = 15x^2 - 4x - 43
\][/tex]
Divide the next leading term:
[tex]\[
\frac{15x^2}{3x} = 5x
\][/tex]
Multiply and subtract:
[tex]\[
(3x - 5) \cdot 5x = 15x^2 - 25x
\][/tex]
[tex]\[
(15x^2 - 4x - 43) - (15x^2 - 25x) = 21x - 43
\][/tex]
Divide the next leading term:
[tex]\[
\frac{21x}{3x} = 7
\][/tex]
Multiply and subtract:
[tex]\[
(3x - 5) \cdot 7 = 21x - 35
\][/tex]
[tex]\[
(21x - 43) - (21x - 35) = -8
\][/tex]
Final Result
The quotient is [tex]\(x^4 - 6x^3 - 5x^2 + 5x + 7\)[/tex] and the remainder is [tex]\(-8\)[/tex].
So, the division of [tex]\(3x^5 - 23x^4 + 15x^3 + 40x^2 - 4x - 43\)[/tex] by [tex]\(3x - 5\)[/tex] results in:
[tex]\[
\text{Quotient: } x^4 - 6x^3 - 5x^2 + 5x + 7, \quad \text{Remainder: } -8
\][/tex]
