To solve the given polynomial division problem, we'll perform polynomial long division step by step. Here's how we can divide \(10x^3 + 16x + 3\) by \(5x^2 + 4\):
1. Setup the division:
- Dividend: \(10x^3 + 16x + 3\)
- Divisor: \(5x^2 + 4\)
2. Divide the leading terms:
- Leading term of the dividend: \(10x^3\)
- Leading term of the divisor: \(5x^2\)
- Divide \(10x^3\) by \(5x^2\):
[tex]\[
\frac{10x^3}{5x^2} = 2x
\][/tex]
- This gives us the first term of the quotient: \(2x\).
3. Multiply the divisor by the first term of the quotient:
- Multiply \(5x^2 + 4\) by \(2x\):
[tex]\[
(5x^2 + 4) \cdot 2x = 10x^3 + 8x
\][/tex]
4. Subtract this result from the original dividend:
- Subtract \(10x^3 + 8x\) from \(10x^3 + 16x + 3\):
[tex]\[
(10x^3 + 16x + 3) - (10x^3 + 8x) = 8x + 3
\][/tex]
5. The result of the subtraction:
- New polynomial after subtraction: \(8x + 3\).
6. Determine if further division is possible:
- Now we need to divide \(8x + 3\) by \(5x^2 + 4\).
- The degree of the remainder polynomial \(8x + 3\) is lower than the degree of the divisor \(5x^2 + 4\), so we stop here.
7. Final result:
- Quotient: \(2x\)
- Remainder: \(8x + 3\)
8. Express the final answer as required:
- Quotient plus the remainder over the divisor:
[tex]\[
\frac{10x^3 + 16x + 3}{5x^2 + 4} = 2x + \frac{8x + 3}{5x^2 + 4}
\][/tex]
Thus, the final answer is:
[tex]\[
\frac{10x^3 + 16x + 3}{5x^2 + 4} = 2x + \frac{8x + 3}{5x^2 + 4}
\][/tex]
