Answer :
To divide the polynomial [tex]\(4x^4 - 24x^3 + 21x^2 - 6x + 5\)[/tex] by [tex]\(x - 5\)[/tex], we'll use polynomial long division. Here is a step-by-step solution:
### Step 1: Set up the division
Write the dividend ([tex]\(4x^4 - 24x^3 + 21x^2 - 6x + 5\)[/tex]) under the division symbol and the divisor ([tex]\(x - 5\)[/tex]) outside of it.
### Step 2: Divide the first term
Divide the first term of the dividend ([tex]\(4x^4\)[/tex]) by the first term of the divisor ([tex]\(x\)[/tex]):
[tex]\[ \frac{4x^4}{x} = 4x^3 \][/tex]
Write [tex]\(4x^3\)[/tex] as the first term of the quotient.
### Step 3: Multiply and subtract
Multiply [tex]\(4x^3\)[/tex] by the divisor ([tex]\(x - 5\)[/tex]):
[tex]\[ 4x^3 \cdot (x - 5) = 4x^4 - 20x^3 \][/tex]
Subtract this from the current dividend:
[tex]\[ (4x^4 - 24x^3) - (4x^4 - 20x^3) = -4x^3 \][/tex]
Bring down the next term from the original dividend, which is [tex]\(21x^2\)[/tex], to get [tex]\(-4x^3 + 21x^2\)[/tex].
### Step 4: Repeat the process
Divide [tex]\(-4x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{-4x^3}{x} = -4x^2 \][/tex]
Write [tex]\(-4x^2\)[/tex] as the next term of the quotient.
Multiply [tex]\(-4x^2\)[/tex] by [tex]\(x - 5\)[/tex]:
[tex]\[ -4x^2 \cdot (x - 5) = -4x^3 + 20x^2 \][/tex]
Subtract:
[tex]\[ (-4x^3 + 21x^2) - (-4x^3 + 20x^2) = x^2 \][/tex]
Bring down the next term, [tex]\(-6x\)[/tex], to get [tex]\(x^2 - 6x\)[/tex].
### Step 5: Continue the division
Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{x^2}{x} = x \][/tex]
Write [tex]\(x\)[/tex] as the next term of the quotient.
Multiply [tex]\(x\)[/tex] by [tex]\(x - 5\)[/tex]:
[tex]\[ x \cdot (x - 5) = x^2 - 5x \][/tex]
Subtract:
[tex]\[ (x^2 - 6x) - (x^2 - 5x) = -x \][/tex]
Bring down the last term, [tex]\(+5\)[/tex], to get [tex]\(-x + 5\)[/tex].
### Step 6: Final step
Divide [tex]\(-x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{-x}{x} = -1 \][/tex]
Write [tex]\(-1\)[/tex] as the last term of the quotient.
Multiply [tex]\(-1\)[/tex] by [tex]\(x - 5\)[/tex]:
[tex]\[ -1 \cdot (x - 5) = -x + 5 \][/tex]
Subtract:
[tex]\[ (-x + 5) - (-x + 5) = 0 \][/tex]
There is no remainder.
### Conclusion
The quotient of the division is:
[tex]\[ 4x^3 - 4x^2 + x - 1 \][/tex]
And since there's no remainder, the division is complete. Therefore, the answer is:
[tex]\[ 4x^3 - 4x^2 + x - 1 \][/tex]
### Step 1: Set up the division
Write the dividend ([tex]\(4x^4 - 24x^3 + 21x^2 - 6x + 5\)[/tex]) under the division symbol and the divisor ([tex]\(x - 5\)[/tex]) outside of it.
### Step 2: Divide the first term
Divide the first term of the dividend ([tex]\(4x^4\)[/tex]) by the first term of the divisor ([tex]\(x\)[/tex]):
[tex]\[ \frac{4x^4}{x} = 4x^3 \][/tex]
Write [tex]\(4x^3\)[/tex] as the first term of the quotient.
### Step 3: Multiply and subtract
Multiply [tex]\(4x^3\)[/tex] by the divisor ([tex]\(x - 5\)[/tex]):
[tex]\[ 4x^3 \cdot (x - 5) = 4x^4 - 20x^3 \][/tex]
Subtract this from the current dividend:
[tex]\[ (4x^4 - 24x^3) - (4x^4 - 20x^3) = -4x^3 \][/tex]
Bring down the next term from the original dividend, which is [tex]\(21x^2\)[/tex], to get [tex]\(-4x^3 + 21x^2\)[/tex].
### Step 4: Repeat the process
Divide [tex]\(-4x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{-4x^3}{x} = -4x^2 \][/tex]
Write [tex]\(-4x^2\)[/tex] as the next term of the quotient.
Multiply [tex]\(-4x^2\)[/tex] by [tex]\(x - 5\)[/tex]:
[tex]\[ -4x^2 \cdot (x - 5) = -4x^3 + 20x^2 \][/tex]
Subtract:
[tex]\[ (-4x^3 + 21x^2) - (-4x^3 + 20x^2) = x^2 \][/tex]
Bring down the next term, [tex]\(-6x\)[/tex], to get [tex]\(x^2 - 6x\)[/tex].
### Step 5: Continue the division
Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{x^2}{x} = x \][/tex]
Write [tex]\(x\)[/tex] as the next term of the quotient.
Multiply [tex]\(x\)[/tex] by [tex]\(x - 5\)[/tex]:
[tex]\[ x \cdot (x - 5) = x^2 - 5x \][/tex]
Subtract:
[tex]\[ (x^2 - 6x) - (x^2 - 5x) = -x \][/tex]
Bring down the last term, [tex]\(+5\)[/tex], to get [tex]\(-x + 5\)[/tex].
### Step 6: Final step
Divide [tex]\(-x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{-x}{x} = -1 \][/tex]
Write [tex]\(-1\)[/tex] as the last term of the quotient.
Multiply [tex]\(-1\)[/tex] by [tex]\(x - 5\)[/tex]:
[tex]\[ -1 \cdot (x - 5) = -x + 5 \][/tex]
Subtract:
[tex]\[ (-x + 5) - (-x + 5) = 0 \][/tex]
There is no remainder.
### Conclusion
The quotient of the division is:
[tex]\[ 4x^3 - 4x^2 + x - 1 \][/tex]
And since there's no remainder, the division is complete. Therefore, the answer is:
[tex]\[ 4x^3 - 4x^2 + x - 1 \][/tex]
