Answer :
When dividing [tex]\(10x^4 + 19x^3 - 32x^2 - 4x - 3\)[/tex] by [tex]\(5x^2 + 3x - 1\)[/tex], the quotient is [tex]\(2x^2 + 2x - 1\)[/tex] with no remainder.
To divide [tex]\(10x^4 + 19x^3 - 32x^2 - 4x - 3\)[/tex] by [tex]\(5x^2 + 3x - 1\)[/tex], we perform polynomial long division. Here's how:
1. Divide the highest-degree term of the dividend by the highest-degree term of the divisor to get the first term of the quotient.
- [tex]\(10x^4 \div 5x^2 = 2x^2\)[/tex].
2. Multiply the entire divisor by the first term of the quotient and subtract it from the dividend.
- [tex]\((2x^2)(5x^2 + 3x - 1) = 10x^4 + 6x^3 - 2x^2\)[/tex].
- Subtracting from the original dividend gives [tex]\(19x^3 - 32x^2 - 4x - 3 - (10x^4 + 6x^3 - 2x^2) = 13x^3 - 30x^2 - 4x - 3\).[/tex]
3. Repeat the process with the new dividend.
- Divide [tex]\(13x^3\) by \(5x^2\)[/tex] to get 2x.
- Multiply and subtract to get [tex]\(13x^3 - 10x^2 - 6x\)[/tex].
4. Repeat until the degree of the remainder is less than the degree of the divisor.
The quotient is [tex]\(2x^2 + 2x - 1\)[/tex] with a remainder of 0.
