Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use long division to divide. $$\left(4 x^{3}-7 x^{2}-11 x+5\right) \div(4 x+5)$$

Step-by-Step Solution

Answer

The solution to the problem is \(x^{2}-\frac{3}{4}x-\frac{1}{16} + \frac{ \frac{21}{16}}{4x+5}\).

Step 1: Setup the Long Division

To set up the long division, write the dividend \(4x^{3}-7x^{2}-11x+5\) inside the division symbol and the divisor \(4x+5\) outside the division symbol, similar to traditional long division.

Step 2: Divide the First Term

Divide the first term of the dividend \(4x^3\) by the first term of the divisor \(4x\). This gives us \(x^2\), which is the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the divisor \(4x+5\) by the first term of the quotient \(x^2\), then subtract this from the original dividend. This gives a new expression of \(-3x^{2}-11x+5\).

Step 4: Repeat the Process

Repeat step 2 with the first term of the new expression and the first term of the divisor. So, \(-3x^2\) divided by \(4x\) gives \(-\frac{3}{4}x\) which is added to the quotient. Substract to get a new expression \( -\frac{1}{4}x +5\).

Step 5: Repeat the Process Again

Repeat step 2 again with the first term of the new expression and the first term of the divisor. So, \(-\frac{1}{4}x\) divided by \(4x\) gives \(-\frac{1}{16}\). After subtraction, we get \( \frac{21}{16}\). Since the degree of this term is less than the degree of the divisor, the division process stops here.

Step 6: Write the Final Result

Combine all terms of the quotient and the remainder to express the final result of the division. The quotient \(x^{2}-\frac{3}{4}x-\frac{1}{16}\) and the remainder \( \frac{21}{16}\) , and the divisor \(4x+5\) makes the final result as \(x^{2}-\frac{3}{4}x-\frac{1}{16} + \frac{ \frac{21}{16}}{4x+5}\).