Question

simplify the following expression: (\frac{21x^{3}-2x^{2}+20x + 11}{7x + 4})
1 (3x^{2}-2x + 4+\frac{27}{7x + 4})
2 (3x^{2}-2x + 4-\frac{5}{7x + 4})
3 (3x^{2}-2x-4-\frac{5}{7x + 4})
4 (3x^{2}-2x + 4-\frac{27}{7x + 4})

Explanation:

Step1: Perform Polynomial Long Division

Divide \(21x^3 - 2x^2 + 20x + 11\) by \(7x + 4\).
First term: \(\frac{21x^3}{7x} = 3x^2\). Multiply \(7x + 4\) by \(3x^2\): \(21x^3 + 12x^2\).
Subtract from dividend: \((21x^3 - 2x^2 + 20x + 11) - (21x^3 + 12x^2) = -14x^2 + 20x + 11\).

Step2: Next Term in Division

Second term: \(\frac{-14x^2}{7x} = -2x\). Multiply \(7x + 4\) by \(-2x\): \(-14x^2 - 8x\).
Subtract: \((-14x^2 + 20x + 11) - (-14x^2 - 8x) = 28x + 11\).

Step3: Next Term in Division

Third term: \(\frac{28x}{7x} = 4\). Multiply \(7x + 4\) by \(4\): \(28x + 16\).
Subtract: \((28x + 11) - (28x + 16) = -5\).

Step4: Write the Result

The division gives \(3x^2 - 2x + 4\) with a remainder of \(-5\). So, \(\frac{21x^3 - 2x^2 + 20x + 11}{7x + 4} = 3x^2 - 2x + 4 - \frac{5}{7x + 4}\).

Answer:

1. \(3x^2 - 2x + 4 - \frac{5}{7x + 4}\)