Divide the polynomial 4x4+4x−10 by 2x2−3 to determine the quotient and remainder. (1 point) Responses The quotient is 2x2−3 and the remainder is 4x+1 The quotient is 2 x squared minus 3 and the remainder is 4x+1 The quotient is 2x2+3 and the remainder is 4x-19 The quotient is 2 x squared plus 3 and the remainder is 4x-19 The quotient is 2x2+3 and the remainder is 4x-1 The quotient is 2 x squared plus 3 and the remainder is 4x-1 The quotient is 2x2−3 and the remainder is 4x-19

Question

GPT-4o mini · Answer

To divide the polynomial $4x^4 + 4x - 10$ by $2x^2 - 3$, we can use polynomial long division. 1. **Set up the long division**: Divide the leading term of the dividend $4x^4$ by the leading term of the divisor $2x^2$: $$ \frac{4x^4}{2x^2} = 2x^2 $$ 2. **Multiply the entire divisor by $2x^2$**: $$ 2x^2(2x^2 - 3) = 4x^4 - 6x^2 $$ 3. **Subtract this from the original polynomial**: $$ (4x^4 + 4x - 10) - (4x^4 - 6x^2) = 6x^2 + 4x - 10 $$ 4. **Repeat the process for $6x^2 + 4x - 10$**: Divide the leading term $6x^2$ by the leading term $2x^2$: $$ \frac{6x^2}{2x^2} = 3 $$ 5. **Multiply the entire divisor by $3$**: $$ 3(2x^2 - 3) = 6x^2 - 9 $$ 6. **Subtract this from $6x^2 + 4x - 10$**: $$ (6x^2 + 4x - 10) - (6x^2 - 9) = 4x - 1 $$ Now, we can write the result of the division: $$ 4x^4 + 4x - 10 = (2x^2 + 3)(2x^2 - 3) + (4x - 1) $$ Thus, the quotient is $2x^2 + 3$ and the remainder is $4x - 1$. So the correct response is: **The quotient is $2x^2 + 3$ and the remainder is $4x - 1$**.