Answer :
To find the quotient when dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x-5\)[/tex], we use polynomial division. Here are the steps:
1. Setup: Write down the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x-5\)[/tex].
2. Divide the Leading Terms: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], which gives [tex]\(x^3\)[/tex]. This is the first term of the quotient.
3. Multiply and Subtract:
- Multiply [tex]\(x^3\)[/tex] by the entire divisor [tex]\(x-5\)[/tex], resulting in [tex]\(x^4 - 5x^3\)[/tex].
- Subtract this from the original dividend: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 5x^3) = 10x^3 - 3x - 15\)[/tex].
4. Repeat the Division Process:
- Divide the new leading term [tex]\(10x^3\)[/tex] by [tex]\(x\)[/tex], yielding [tex]\(10x^2\)[/tex].
- Multiply [tex]\(10x^2\)[/tex] by [tex]\(x-5\)[/tex] to get [tex]\(10x^3 - 50x^2\)[/tex].
- Subtract: [tex]\((10x^3 - 3x - 15) - (10x^3 - 50x^2) = 50x^2 - 3x - 15\)[/tex].
5. Continue the Division:
- Divide [tex]\(50x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(50x\)[/tex].
- Multiply [tex]\(50x\)[/tex] by [tex]\(x-5\)[/tex], giving [tex]\(50x^2 - 250x\)[/tex].
- Subtract: [tex]\((50x^2 - 3x - 15) - (50x^2 - 250x) = 247x - 15\)[/tex].
6. Final Step:
- Divide [tex]\(247x\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(247\)[/tex].
- Multiply [tex]\(247\)[/tex] by [tex]\(x-5\)[/tex], yielding [tex]\(247x - 1235\)[/tex].
- Subtracting these gives the remainder: [tex]\((247x - 15) - (247x - 1235) = 1220\)[/tex].
The quotient is [tex]\((x^3 + 10x^2 + 50x + 247)\)[/tex], and the remainder is [tex]\(1220\)[/tex].
1. Setup: Write down the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x-5\)[/tex].
2. Divide the Leading Terms: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], which gives [tex]\(x^3\)[/tex]. This is the first term of the quotient.
3. Multiply and Subtract:
- Multiply [tex]\(x^3\)[/tex] by the entire divisor [tex]\(x-5\)[/tex], resulting in [tex]\(x^4 - 5x^3\)[/tex].
- Subtract this from the original dividend: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 5x^3) = 10x^3 - 3x - 15\)[/tex].
4. Repeat the Division Process:
- Divide the new leading term [tex]\(10x^3\)[/tex] by [tex]\(x\)[/tex], yielding [tex]\(10x^2\)[/tex].
- Multiply [tex]\(10x^2\)[/tex] by [tex]\(x-5\)[/tex] to get [tex]\(10x^3 - 50x^2\)[/tex].
- Subtract: [tex]\((10x^3 - 3x - 15) - (10x^3 - 50x^2) = 50x^2 - 3x - 15\)[/tex].
5. Continue the Division:
- Divide [tex]\(50x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(50x\)[/tex].
- Multiply [tex]\(50x\)[/tex] by [tex]\(x-5\)[/tex], giving [tex]\(50x^2 - 250x\)[/tex].
- Subtract: [tex]\((50x^2 - 3x - 15) - (50x^2 - 250x) = 247x - 15\)[/tex].
6. Final Step:
- Divide [tex]\(247x\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(247\)[/tex].
- Multiply [tex]\(247\)[/tex] by [tex]\(x-5\)[/tex], yielding [tex]\(247x - 1235\)[/tex].
- Subtracting these gives the remainder: [tex]\((247x - 15) - (247x - 1235) = 1220\)[/tex].
The quotient is [tex]\((x^3 + 10x^2 + 50x + 247)\)[/tex], and the remainder is [tex]\(1220\)[/tex].
