Answer :
Sure! Let's perform the polynomial division step by step to divide [tex]\((5x^4 - 2x^3 - 21x^2 + 0x + 13)\)[/tex] by [tex]\((x - 2)\)[/tex].
### Step-by-Step Solution:
1. Identify the Terms:
- Dividend (Numerator): [tex]\(5x^4 - 2x^3 - 21x^2 + 0x + 13\)[/tex]
- Divisor: [tex]\(x - 2\)[/tex]
2. Set Up Division:
- The goal is to divide each component step by step, focusing on leading terms.
3. First Division:
- Divide [tex]\(5x^4\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(5x^3\)[/tex].
- Multiply [tex]\(5x^3\)[/tex] by the whole divisor [tex]\((x - 2)\)[/tex] to get [tex]\(5x^4 - 10x^3\)[/tex].
- Subtract from the original polynomial:
[tex]\[
(5x^4 - 2x^3) - (5x^4 - 10x^3) = 8x^3
\][/tex]
4. Second Division:
- Bring down the next term: [tex]\(8x^3 - 21x^2\)[/tex].
- Divide [tex]\(8x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(8x^2\)[/tex].
- Multiply [tex]\(8x^2\)[/tex] by [tex]\((x - 2)\)[/tex] to get [tex]\(8x^3 - 16x^2\)[/tex].
- Subtract:
[tex]\[
(8x^3 - 21x^2) - (8x^3 - 16x^2) = -5x^2
\][/tex]
5. Third Division:
- Bring down the next term: [tex]\(-5x^2 + 0x\)[/tex].
- Divide [tex]\(-5x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-5x\)[/tex].
- Multiply [tex]\(-5x\)[/tex] by [tex]\((x - 2)\)[/tex] to get [tex]\(-5x^2 + 10x\)[/tex].
- Subtract:
[tex]\[
(-5x^2 + 0x) - (-5x^2 + 10x) = -10x
\][/tex]
6. Fourth Division:
- Bring down the next term: [tex]\(-10x + 13\)[/tex].
- Divide [tex]\(-10x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-10\)[/tex].
- Multiply [tex]\(-10\)[/tex] by [tex]\((x - 2)\)[/tex] to get [tex]\(-10x + 20\)[/tex].
- Subtract:
[tex]\[
(-10x + 13) - (-10x + 20) = -7
\][/tex]
7. Conclusion:
- The quotient is [tex]\(5x^3 + 8x^2 - 5x - 10\)[/tex].
- The remainder is [tex]\(-7\)[/tex].
Thus, the result of the division [tex]\((5x^4 - 2x^3 - 21x^2 + 13) \div (x - 2)\)[/tex] is:
[tex]\[ \text{Quotient: } 5x^3 + 8x^2 - 5x - 10 \][/tex]
[tex]\[ \text{Remainder: } -7 \][/tex]
### Step-by-Step Solution:
1. Identify the Terms:
- Dividend (Numerator): [tex]\(5x^4 - 2x^3 - 21x^2 + 0x + 13\)[/tex]
- Divisor: [tex]\(x - 2\)[/tex]
2. Set Up Division:
- The goal is to divide each component step by step, focusing on leading terms.
3. First Division:
- Divide [tex]\(5x^4\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(5x^3\)[/tex].
- Multiply [tex]\(5x^3\)[/tex] by the whole divisor [tex]\((x - 2)\)[/tex] to get [tex]\(5x^4 - 10x^3\)[/tex].
- Subtract from the original polynomial:
[tex]\[
(5x^4 - 2x^3) - (5x^4 - 10x^3) = 8x^3
\][/tex]
4. Second Division:
- Bring down the next term: [tex]\(8x^3 - 21x^2\)[/tex].
- Divide [tex]\(8x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(8x^2\)[/tex].
- Multiply [tex]\(8x^2\)[/tex] by [tex]\((x - 2)\)[/tex] to get [tex]\(8x^3 - 16x^2\)[/tex].
- Subtract:
[tex]\[
(8x^3 - 21x^2) - (8x^3 - 16x^2) = -5x^2
\][/tex]
5. Third Division:
- Bring down the next term: [tex]\(-5x^2 + 0x\)[/tex].
- Divide [tex]\(-5x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-5x\)[/tex].
- Multiply [tex]\(-5x\)[/tex] by [tex]\((x - 2)\)[/tex] to get [tex]\(-5x^2 + 10x\)[/tex].
- Subtract:
[tex]\[
(-5x^2 + 0x) - (-5x^2 + 10x) = -10x
\][/tex]
6. Fourth Division:
- Bring down the next term: [tex]\(-10x + 13\)[/tex].
- Divide [tex]\(-10x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-10\)[/tex].
- Multiply [tex]\(-10\)[/tex] by [tex]\((x - 2)\)[/tex] to get [tex]\(-10x + 20\)[/tex].
- Subtract:
[tex]\[
(-10x + 13) - (-10x + 20) = -7
\][/tex]
7. Conclusion:
- The quotient is [tex]\(5x^3 + 8x^2 - 5x - 10\)[/tex].
- The remainder is [tex]\(-7\)[/tex].
Thus, the result of the division [tex]\((5x^4 - 2x^3 - 21x^2 + 13) \div (x - 2)\)[/tex] is:
[tex]\[ \text{Quotient: } 5x^3 + 8x^2 - 5x - 10 \][/tex]
[tex]\[ \text{Remainder: } -7 \][/tex]
