Questions: In the following long division problem, the first step has been completed. 2x - 4 divides into 14x^2 - 26x + 18 Fill in each blank so that the resulting statement is true. The next step is to multiply by 7x and 2x. The result is. Write this result below.

Solution Steps

To solve this problem, we need to perform polynomial long division. The first step involves dividing the leading term of the dividend by the leading term of the divisor to find the first term of the quotient. Then, multiply the entire divisor by this term and subtract the result from the original polynomial. Repeat this process with the new polynomial formed after subtraction until the degree of the remainder is less than the degree of the divisor.

To begin the polynomial long division, divide the leading term of the dividend, \(14x^2\), by the leading term of the divisor, \(2x\). This gives us the first term of the quotient: \[ \frac{14x^2}{2x} = 7x \]

Multiply the entire divisor, \(2x - 4\), by the term obtained in Step 1, \(7x\): \[ 7x \cdot (2x - 4) = 14x^2 - 28x \] Subtract this result from the original dividend: \[ (14x^2 - 26x + 18) - (14x^2 - 28x) = 2x + 18 \]

Now, divide the new leading term, \(2x\), by the leading term of the divisor, \(2x\): \[ \frac{2x}{2x} = 1 \] Multiply the entire divisor by this new term: \[ 1 \cdot (2x - 4) = 2x - 4 \] Subtract this from the current polynomial: \[ (2x + 18) - (2x - 4) = 22 \]

Final Answer