In Exercises 17–32, divide using synthetic division. (x^7+x^5−10x^3+12)/(x+2)

Verified step by step guidance

1. Identify the divisor and the dividend. The divisor is \(x + 2\), and the dividend is \(x^7 + x^5 - 10x^3 + 12\).
2. Set up synthetic division by writing the zero of the divisor, which is \(-2\), on the left. Write the coefficients of the dividend in descending order of power: \(1, 0, 1, 0, -10, 0, 0, 12\).
3. Bring down the leading coefficient (1) to the bottom row.
4. Multiply the number you just brought down (1) by \(-2\) and write the result under the next coefficient (0). Add this result to the coefficient above it and write the sum below.
5. Repeat the multiplication and addition process for each coefficient until you reach the end of the row.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing polynomials, 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 than traditional long division. This technique streamlines calculations and is especially advantageous for higher-degree polynomials.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this context, the polynomial is represented as x^7 + x^5 - 10x^3 + 12, where the highest power indicates the degree of the polynomial. Understanding polynomial functions is crucial for performing operations like division, as it helps in identifying the structure and behavior of the function.

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 is useful in synthetic division as it allows for quick verification of the results. In the given problem, since we are dividing by (x + 2), we can evaluate the polynomial at x = -2 to find the remainder after division.