The image shows the first few steps of the polynomial division (3x^4+9x^2−13)÷(x^2−5x) . Complete the division process and find the remainder of this polynomial division.
1. The remainder is 407x
2. The remainder is 2,087
3. The remainder is 420x-13
4. The remainder is 407
The remainder of the polynomial division (3x^4 + 9x^2 -13) ÷ (x^2 - 5x) is 420x - 13. Therefore, the correct answer is option 3.
To complete the polynomial division (3x^4+9x^2−13) ÷ (x^2−5x), follow these steps:
Step 1: Divide the first term of the dividend (3x^4) by the first term of the divisor (x^2). The result is 3x^2.
Step 2: Multiply the divisor (x^2−5x) by the result of the previous step (3x^2), which gives you 3x^4−15x^3.
Step 3: Subtract the product obtained in the previous step (3x^4−15x^3) from the dividend (3x^4+9x^2−13). This gives you 15x^3+9x^2−13.
Step 4: Bring down the next term from the dividend, which is 0x. You now have 15x^3+9x^2−13+0x.
Step 5: Divide the first term of the new expression (15x^3) by the first term of the divisor (x^2). The result is 15x.
Step 6: Multiply the divisor (x^2−5x) by the result of the previous step (15x), which gives you 15x^3−75x^2.
Step 7: Subtract the product obtained in the previous step (15x^3−75x^2) from the new expression (15x^3+9x^2−13+0x). This gives you 84x^2−13+0x.
Step 8: Divide the first term of the new expression (84x^2) by the first term of the divisor (x^2). The result is 84.
Step 9: Multiply the divisor (x^2−5x) by the result of the previous step (84), which gives you 84x^2−420x.
Step 10: Subtract the product obtained in the previous step (84x^2−420x) from the new expression (84x^2−13+0x). This gives you 420x−13+0x.
Since there are no more terms left in the dividend, the resulting expression after division is 420x−13. Therefore, the remainder of this polynomial division is 420x−13.
Hence, the correct answer is option 3: The remainder is 420x-13.
To complete the polynomial division (3x^4+9x^2−13)÷(x^2−5x), you need to follow these steps:
1. Write the dividend (the polynomial being divided): 3x^4+9x^2−13.
2. Write the divisor (the polynomial you're dividing by): x^2−5x.
3. Divide the highest degree term of the dividend by the highest degree term of the divisor. Here, you'll divide 3x^4 by x^2 to get 3x^2.
4. Multiply the divisor (x^2−5x) by the quotient you just found (3x^2). This gives you 3x^4−15x^3.
5. Subtract this product (3x^4−15x^3) from the original dividend (3x^4+9x^2−13). The result is 24x^3+9x^2−13.
6. Repeat steps 3-5 with the new dividend (24x^3+9x^2−13) and the original divisor (x^2−5x).
7. Divide the highest degree term of the new dividend (24x^3) by the highest degree term of the divisor (x^2), which gives you 24x.
8. Multiply the divisor (x^2−5x) by the new quotient (24x). This gives you 24x^3−120x^2.
9. Subtract this product (24x^3−120x^2) from the new dividend (24x^3+9x^2−13). The result is 129x^2−13.
10. Repeat steps 3-5 with the new dividend (129x^2−13) and the original divisor (x^2−5x).
11. Divide the highest degree term of the new dividend (129x^2) by the highest degree term of the divisor (x^2), which gives you 129.
12. Multiply the divisor (x^2−5x) by the new quotient (129). This gives you 129x^2−645x.
13. Subtract this product (129x^2−645x) from the new dividend (129x^2−13). The result is 632x+13.
14. Since the degree of the new dividend (632x+13) is less than the degree of the divisor (x^2−5x), the division process is complete.
15. The remainder of the polynomial division is 632x+13.
Therefore, the correct answer is:
3. The remainder is 632x+13.