Answer :
To divide the polynomial [tex]\(4x^4 - 23x^3 + 16x^2 - 9x + 20\)[/tex] by [tex]\(x - 5\)[/tex], we can use polynomial division. Here's how you can do it step-by-step:
1. Identify the terms: We're dividing the polynomial [tex]\(4x^4 - 23x^3 + 16x^2 - 9x + 20\)[/tex] by the polynomial [tex]\(x - 5\)[/tex].
2. Divide the first terms: Take the leading term of the dividend, [tex]\(4x^4\)[/tex], and divide it by the leading term of the divisor, [tex]\(x\)[/tex]. This gives [tex]\(4x^3\)[/tex].
3. Multiply and subtract: Multiply [tex]\(4x^3\)[/tex] by the entire divisor [tex]\(x - 5\)[/tex], resulting in [tex]\(4x^4 - 20x^3\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(4x^4 - 23x^3 + 16x^2 - 9x + 20) - (4x^4 - 20x^3) = -3x^3 + 16x^2 - 9x + 20
\][/tex]
4. Repeat the process: Take the new leading term, [tex]\(-3x^3\)[/tex], and divide by [tex]\(x\)[/tex] to get [tex]\(-3x^2\)[/tex]. Multiply [tex]\(-3x^2\)[/tex] by [tex]\(x - 5\)[/tex] giving [tex]\(-3x^3 + 15x^2\)[/tex]. Subtract this from [tex]\(-3x^3 + 16x^2 - 9x + 20\)[/tex]:
[tex]\[
(-3x^3 + 16x^2 - 9x + 20) - (-3x^3 + 15x^2) = x^2 - 9x + 20
\][/tex]
5. Continue: Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x\)[/tex]. Multiply [tex]\(x\)[/tex] by [tex]\(x - 5\)[/tex] resulting in [tex]\(x^2 - 5x\)[/tex]. Subtract:
[tex]\[
(x^2 - 9x + 20) - (x^2 - 5x) = -4x + 20
\][/tex]
6. Last divide: Divide [tex]\(-4x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-4\)[/tex]. Multiply [tex]\(-4\)[/tex] by [tex]\(x - 5\)[/tex] to get [tex]\(-4x + 20\)[/tex]. Subtract:
[tex]\[
(-4x + 20) - (-4x + 20) = 0
\][/tex]
The quotient of the division is [tex]\(4x^3 - 3x^2 + x - 4\)[/tex] and the remainder is [tex]\(0\)[/tex].
So, the final result of the division is:
[tex]\[
\frac{4x^4 - 23x^3 + 16x^2 - 9x + 20}{x - 5} = 4x^3 - 3x^2 + x - 4
\][/tex]
1. Identify the terms: We're dividing the polynomial [tex]\(4x^4 - 23x^3 + 16x^2 - 9x + 20\)[/tex] by the polynomial [tex]\(x - 5\)[/tex].
2. Divide the first terms: Take the leading term of the dividend, [tex]\(4x^4\)[/tex], and divide it by the leading term of the divisor, [tex]\(x\)[/tex]. This gives [tex]\(4x^3\)[/tex].
3. Multiply and subtract: Multiply [tex]\(4x^3\)[/tex] by the entire divisor [tex]\(x - 5\)[/tex], resulting in [tex]\(4x^4 - 20x^3\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(4x^4 - 23x^3 + 16x^2 - 9x + 20) - (4x^4 - 20x^3) = -3x^3 + 16x^2 - 9x + 20
\][/tex]
4. Repeat the process: Take the new leading term, [tex]\(-3x^3\)[/tex], and divide by [tex]\(x\)[/tex] to get [tex]\(-3x^2\)[/tex]. Multiply [tex]\(-3x^2\)[/tex] by [tex]\(x - 5\)[/tex] giving [tex]\(-3x^3 + 15x^2\)[/tex]. Subtract this from [tex]\(-3x^3 + 16x^2 - 9x + 20\)[/tex]:
[tex]\[
(-3x^3 + 16x^2 - 9x + 20) - (-3x^3 + 15x^2) = x^2 - 9x + 20
\][/tex]
5. Continue: Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x\)[/tex]. Multiply [tex]\(x\)[/tex] by [tex]\(x - 5\)[/tex] resulting in [tex]\(x^2 - 5x\)[/tex]. Subtract:
[tex]\[
(x^2 - 9x + 20) - (x^2 - 5x) = -4x + 20
\][/tex]
6. Last divide: Divide [tex]\(-4x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-4\)[/tex]. Multiply [tex]\(-4\)[/tex] by [tex]\(x - 5\)[/tex] to get [tex]\(-4x + 20\)[/tex]. Subtract:
[tex]\[
(-4x + 20) - (-4x + 20) = 0
\][/tex]
The quotient of the division is [tex]\(4x^3 - 3x^2 + x - 4\)[/tex] and the remainder is [tex]\(0\)[/tex].
So, the final result of the division is:
[tex]\[
\frac{4x^4 - 23x^3 + 16x^2 - 9x + 20}{x - 5} = 4x^3 - 3x^2 + x - 4
\][/tex]
