Answer :
To solve the division [tex]\((2x^3 - 19x^2 + 38x + 24) \div (x - 4)\)[/tex], we'll perform polynomial division. Here’s a step-by-step explanation of the process:
### Step 1: Set up the Division
Write the dividend [tex]\(2x^3 - 19x^2 + 38x + 24\)[/tex] under the division bar and the divisor [tex]\(x - 4\)[/tex] outside.
### Step 2: Divide the Leading Terms
- Divide the leading term of the dividend [tex]\(2x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]: [tex]\(\frac{2x^3}{x} = 2x^2\)[/tex].
- Place [tex]\(2x^2\)[/tex] as the first term of the quotient.
### Step 3: Multiply and Subtract
- Multiply [tex]\(2x^2\)[/tex] by [tex]\(x - 4\)[/tex]: [tex]\(2x^2(x - 4) = 2x^3 - 8x^2\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(2x^3 - 19x^2) - (2x^3 - 8x^2) = -11x^2
\][/tex]
- Bring down the next term [tex]\(+38x\)[/tex] to get [tex]\(-11x^2 + 38x\)[/tex].
### Step 4: Repeat the Process
- Divide the leading term [tex]\(-11x^2\)[/tex] by [tex]\(x\)[/tex]: [tex]\(\frac{-11x^2}{x} = -11x\)[/tex].
- Place [tex]\(-11x\)[/tex] as the next term of the quotient.
- Multiply [tex]\(-11x\)[/tex] by [tex]\(x - 4\)[/tex]: [tex]\(-11x(x - 4) = -11x^2 + 44x\)[/tex].
- Subtract this from [tex]\(-11x^2 + 38x\)[/tex]:
[tex]\[
(-11x^2 + 38x) - (-11x^2 + 44x) = -6x
\][/tex]
- Bring down the final term [tex]\(+24\)[/tex] to get [tex]\(-6x + 24\)[/tex].
### Step 5: Final Step
- Divide the leading term [tex]\(-6x\)[/tex] by [tex]\(x\)[/tex]: [tex]\(\frac{-6x}{x} = -6\)[/tex].
- Place [tex]\(-6\)[/tex] as the last term of the quotient.
- Multiply [tex]\(-6\)[/tex] by [tex]\(x - 4\)[/tex]: [tex]\(-6(x - 4) = -6x + 24\)[/tex].
- Subtract this from [tex]\(-6x + 24\)[/tex]:
[tex]\[
(-6x + 24) - (-6x + 24) = 0
\][/tex]
### Conclusion
The division is exact with no remainder. Therefore, the quotient of [tex]\((2x^3 - 19x^2 + 38x + 24) \div (x - 4)\)[/tex] is:
[tex]\[ 2x^2 - 11x - 6 \][/tex]
That's your final answer!
### Step 1: Set up the Division
Write the dividend [tex]\(2x^3 - 19x^2 + 38x + 24\)[/tex] under the division bar and the divisor [tex]\(x - 4\)[/tex] outside.
### Step 2: Divide the Leading Terms
- Divide the leading term of the dividend [tex]\(2x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]: [tex]\(\frac{2x^3}{x} = 2x^2\)[/tex].
- Place [tex]\(2x^2\)[/tex] as the first term of the quotient.
### Step 3: Multiply and Subtract
- Multiply [tex]\(2x^2\)[/tex] by [tex]\(x - 4\)[/tex]: [tex]\(2x^2(x - 4) = 2x^3 - 8x^2\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(2x^3 - 19x^2) - (2x^3 - 8x^2) = -11x^2
\][/tex]
- Bring down the next term [tex]\(+38x\)[/tex] to get [tex]\(-11x^2 + 38x\)[/tex].
### Step 4: Repeat the Process
- Divide the leading term [tex]\(-11x^2\)[/tex] by [tex]\(x\)[/tex]: [tex]\(\frac{-11x^2}{x} = -11x\)[/tex].
- Place [tex]\(-11x\)[/tex] as the next term of the quotient.
- Multiply [tex]\(-11x\)[/tex] by [tex]\(x - 4\)[/tex]: [tex]\(-11x(x - 4) = -11x^2 + 44x\)[/tex].
- Subtract this from [tex]\(-11x^2 + 38x\)[/tex]:
[tex]\[
(-11x^2 + 38x) - (-11x^2 + 44x) = -6x
\][/tex]
- Bring down the final term [tex]\(+24\)[/tex] to get [tex]\(-6x + 24\)[/tex].
### Step 5: Final Step
- Divide the leading term [tex]\(-6x\)[/tex] by [tex]\(x\)[/tex]: [tex]\(\frac{-6x}{x} = -6\)[/tex].
- Place [tex]\(-6\)[/tex] as the last term of the quotient.
- Multiply [tex]\(-6\)[/tex] by [tex]\(x - 4\)[/tex]: [tex]\(-6(x - 4) = -6x + 24\)[/tex].
- Subtract this from [tex]\(-6x + 24\)[/tex]:
[tex]\[
(-6x + 24) - (-6x + 24) = 0
\][/tex]
### Conclusion
The division is exact with no remainder. Therefore, the quotient of [tex]\((2x^3 - 19x^2 + 38x + 24) \div (x - 4)\)[/tex] is:
[tex]\[ 2x^2 - 11x - 6 \][/tex]
That's your final answer!
