Divide the polynomials 4x^4 + 4x - 10 by 2x^2 - 3 to determine the quotient and remainder.
To divide the polynomials 4x^4 + 4x - 10 by 2x^2 - 3, we use long division.
2x^2 + 4
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2x^2 - 3 | 4x^4 + 0x^3 + 0x^2 + 4x - 10
- (4x^4 - 6x^2)
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6x^2 + 4x
- (6x^2 - 9)
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13x - 10
So the quotient is 2x^2 + 4 and the remainder is 13x - 10.
To divide the polynomials 4x^4 + 4x - 10 by 2x^2 - 3, follow these steps:
Step 1: Arrange the polynomials in descending order of exponents:
4x^4 + 0x^3 + 0x^2 + 4x - 10
2x^2 + 0x - 3
Step 2: Divide the first term of the dividend (4x^4) by the first term of the divisor (2x^2), which gives us 2x^2.
Step 3: Multiply the divisor (2x^2) by the result from step 2 (2x^2), which gives us 4x^4.
Step 4: Subtract the result from step 3 (4x^4) from the dividend (4x^4 + 4x - 10), which cancels out the first term. The result is:
0x^4 + 4x - 10
Step 5: Bring down the next term from the dividend (4x) to form a new dividend:
0x^4 + 4x - 10
2x^2 - 3
Step 6: Divide the first term of the new dividend (4x) by the first term of the divisor (2x^2), which gives us 2x.
Step 7: Multiply the divisor (2x^2) by the result from step 6 (2x), which gives us 4x^2.
Step 8: Subtract the result from step 7 (4x^2) from the new dividend (4x^2 + 4x - 10), which cancels out the first term. The result is:
0x^2 + 4x - 10
Step 9: Bring down the next term from the new dividend (-10) to form a new dividend:
0x^2 + 4x - 10
2x^2 - 3
Step 10: Divide the first term of the new dividend (-10) by the first term of the divisor (2x^2), which gives us -5.
Step 11: Multiply the divisor (2x^2) by the result from step 10 (-5), which gives us -10x^2.
Step 12: Subtract the result from step 11 (-10x^2) from the new dividend (0x^2 + 4x - 10), which cancels out the first term. The result is:
0x^2 + 4x - 10 + (-10x^2) = -10x^2 + 4x - 10.
Step 13: We have no more terms to bring down from the dividend, so our division is complete.
Therefore, the quotient is 2x + (-5) and the remainder is -10x^2 + 4x - 10.
To divide the polynomials 4x^4 + 4x - 10 by 2x^2 - 3, we can use long division method. Here's how to do it step by step:
Step 1: Arrange the terms in descending order of powers of x and make sure all powers are present. If necessary, add zero placeholders for missing terms. In this case, since the degree of the dividend is higher than the degree of the divisor, we don't need to add any zero placeholders.
Step 2: Divide the first term of the dividend (4x^4) by the first term of the divisor (2x^2). This gives us 2x^2.
Step 3: Multiply the entire divisor (2x^2 - 3) by the quotient from step 2 (2x^2). This gives us 4x^4 - 6x^2.
Step 4: Subtract the product obtained in step 3 (4x^4 - 6x^2) from the original dividend (4x^4 + 4x - 10). This gives us (4x^4 + 4x - 10) - (4x^4 - 6x^2).
Simplifying this, we get 10x^2 + 4x - 10.
Step 5: Bring down the next term from the original dividend (-10x^2 + 4x - 10). Now we have 10x^2 + 4x - 10 as the new dividend.
Step 6: Repeat steps 2-5 until we can no longer divide.
Continuing the division process, we have:
Step 2: Divide the first term of the new dividend (10x^2) by the first term of the divisor (2x^2). This gives us 5.
Step 3: Multiply the entire divisor (2x^2 - 3) by the quotient from step 2 (5). This gives us 10x^2 - 15.
Step 4: Subtract the product obtained in step 3 (10x^2 - 15) from the new dividend (10x^2 + 4x - 10). This gives us (10x^2 + 4x - 10) - (10x^2 - 15).
Simplifying this, we get 19x - 5.
Step 5: Bring down the next term from the new dividend (19x - 5). Now we have 19x - 5 as the new dividend.
Step 6: Since the degree of the new dividend (19x - 5) is less than the degree of the divisor (2x^2 - 3), we cannot divide further.
Therefore, the quotient is 2x^2 + 5 and the remainder is 19x - 5.
In summary, when we divide the polynomial 4x^4 + 4x - 10 by 2x^2 - 3, the quotient is 2x^2 + 5 and the remainder is 19x - 5.