Questions: (20 x^3-14 x^2-19 x+5) ÷ (5 x+4) (20 x^3-14 x^2-19 x+5) ÷ (5 x+4) = □

(20 x^3-14 x^2-19 x+5) ÷ (5 x+4) (20 x^3-14 x^2-19 x+5) ÷ (5 x+4) = □
Transcript text: \[ \left(20 x^{3}-14 x^{2}-19 x+5\right) \div(5 x+4) \\ \left(20 x^{3}-14 x^{2}-19 x+5\right) \div(5 x+4)=\square \]
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Solution

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Solution Steps

To divide the polynomial \(20x^3 - 14x^2 - 19x + 5\) by \(5x + 4\), we can use polynomial long division. The process involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by this result, subtracting from the original polynomial, and repeating the process with the remainder until the degree of the remainder is less than the degree of the divisor.

Step 1: Set Up the Division

We are tasked with dividing the polynomial \(20x^3 - 14x^2 - 19x + 5\) by \(5x + 4\). We will perform polynomial long division to find the quotient and remainder.

Step 2: Perform the Division
  1. Divide the leading term of the dividend \(20x^3\) by the leading term of the divisor \(5x\): \[ \frac{20x^3}{5x} = 4x^2 \]
  2. Multiply the entire divisor \(5x + 4\) by \(4x^2\): \[ 4x^2(5x + 4) = 20x^3 + 16x^2 \]
  3. Subtract this result from the original polynomial: \[ (20x^3 - 14x^2 - 19x + 5) - (20x^3 + 16x^2) = -30x^2 - 19x + 5 \]
Step 3: Continue the Division
  1. Now, divide the leading term \(-30x^2\) by \(5x\): \[ \frac{-30x^2}{5x} = -6x \]
  2. Multiply the divisor by \(-6x\): \[ -6x(5x + 4) = -30x^2 - 24x \]
  3. Subtract this from the current polynomial: \[ (-30x^2 - 19x + 5) - (-30x^2 - 24x) = 5x + 5 \]
Step 4: Final Division Step
  1. Divide the leading term \(5x\) by \(5x\): \[ \frac{5x}{5x} = 1 \]
  2. Multiply the divisor by \(1\): \[ 1(5x + 4) = 5x + 4 \]
  3. Subtract this from the current polynomial: \[ (5x + 5) - (5x + 4) = 1 \]

Final Answer

The quotient of the division is \(4x^2 - 6x + 1\) and the remainder is \(1\). Thus, we can express the result of the division as: \[ \boxed{4x^2 - 6x + 1 \quad \text{with a remainder of } 1} \]

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