Perform each division. See Examples 9 and 10. (4x^3-3x^2+1)/(x-2)

Verified step by step guidance

1. Identify the dividend and the divisor. Here, the dividend is \(4x^3 - 3x^2 + 1\) and the divisor is \(x - 2\).
2. Set up the division using polynomial long division. Write \(4x^3 - 3x^2 + 0x + 1\) under the division symbol and \(x - 2\) outside.
3. Divide the first term of the dividend \(4x^3\) by the first term of the divisor \(x\) to get the first term of the quotient, which is \(4x^2\).
4. Multiply the entire divisor \(x - 2\) by \(4x^2\) and subtract the result from the current dividend to find the new dividend.
5. Repeat the process: divide the new leading term of the dividend by the leading term of the divisor, multiply, and subtract, until the degree of the remainder is less than the degree of the divisor.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide a polynomial by another polynomial of lower degree, similar to long division with numbers. The process involves determining how many times the divisor can fit into the leading term of the dividend, subtracting the result, and repeating the process with the remainder until the degree of the remainder is less than that of the divisor.

Synthetic Division

Synthetic division is a simplified form of polynomial division that is particularly useful when dividing by linear factors. It involves using the coefficients of the polynomial and a specific value (the root of the divisor) to perform the division in a more efficient manner, allowing for quicker calculations without the need for writing out all terms.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is equal to f(c). This theorem provides a quick way to evaluate the remainder without performing the entire division process, and it can also help in determining factors of the polynomial.