Questions: (15x^3+22x^2-20x+3)/(5x-1)

Transcript text: $\frac{15 x^{3}+22 x^{2}-20 x+3}{5 x-1}$

Solution

Solution Steps

To solve the problem of dividing the polynomial $ \frac{15x^3 + 22x^2 - 20x + 3}{5x - 1} $, we can use polynomial long division. The goal is to divide the numerator by the denominator to find the quotient and remainder.

Step 1: Polynomial Long Division

We start with the polynomial division of $ 15x^3 + 22x^2 - 20x + 3 $ by $ 5x - 1 $.

Step 2: Finding the Quotient

The first term of the quotient is obtained by dividing the leading term of the numerator $ 15x^3 $ by the leading term of the denominator $ 5x $: \[ \frac{15x^3}{5x} = 3x^2 \] We multiply $ 3x^2 $ by the entire denominator $ 5x - 1 $: \[ 3x^2(5x - 1) = 15x^3 - 3x^2 \] Subtracting this from the original polynomial gives: \[ (15x^3 + 22x^2 - 20x + 3) - (15x^3 - 3x^2) = 25x^2 - 20x + 3 \]

Step 3: Continuing the Division

Next, we repeat the process with the new polynomial $ 25x^2 - 20x + 3 $. The next term of the quotient is: \[ \frac{25x^2}{5x} = 5x \] Multiplying $ 5x $ by the denominator: \[ 5x(5x - 1) = 25x^2 - 5x \] Subtracting this from $ 25x^2 - 20x + 3 $ gives: \[ (25x^2 - 20x + 3) - (25x^2 - 5x) = -15x + 3 \]

Step 4: Final Division Step

Now we divide $ -15x + 3 $ by $ 5x - 1 $: \[ \frac{-15x}{5x} = -3 \] Multiplying $ -3 $ by the denominator: \[ -3(5x - 1) = -15x + 3 \] Subtracting this from $ -15x + 3 $ results in: \[ (-15x + 3) - (-15x + 3) = 0 \] Thus, there is no remainder.

Final Answer

The result of the polynomial division is: \[ \frac{15x^3 + 22x^2 - 20x + 3}{5x - 1} = 3x^2 + 5x - 3 \] The final answer is: \[ \boxed{3x^2 + 5x - 3} \]

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