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STEP 1

1. We are using synthetic division to divide polynomials.
2. The divisor is always a binomial of the form $x - c$ .
3. The dividend is a polynomial that may have missing terms, which need to be accounted for with zero coefficients.

STEP 2

1. Set up the synthetic division for each example.
2. Perform the synthetic division process.
3. Write the resulting quotient and remainder.

STEP 3

For each example, identify the coefficients of the dividend polynomial and the root of the divisor binomial. If any powers are missing in the dividend, use 0 as the coefficient for that term.
Example 1: $(x^2 + 3x - 43) \div (x + 8)$ - Coefficients: $1, 3, -43$ - Root of divisor: $-8$
Example 2: $(2n^2 - 16n + 19) \div (n - 7)$ - Coefficients: $2, -16, 19$ - Root of divisor: $7$
Example 3: $(y^2 + 5y^2 - 7y + 2) \div (y - 2)$ - Coefficients: $1, 5, -7, 2$ - Root of divisor: $2$
Example 4: $(c^4 + 9c^3 + 13c^2 - 33c - 16) \div (c + 4)$ - Coefficients: $1, 9, 13, -33, -16$ - Root of divisor: $-4$
Example 5: $(3k^2 - 17k - 32) \div (k + 5)$ - Coefficients: $3, -17, -32$ - Root of divisor: $-5$
Example 6: $(8x^4 + 64x^3 + 3x - 28) \div (x - 8)$ - Coefficients: $8, 64, 0, 3, -28$ - Root of divisor: $8$

STEP 4

Perform synthetic division for each example using the identified coefficients and root of the divisor.
Example 1: - Setup: $[-8 | 1, 3, -43]$ - Process: Bring down the first coefficient, multiply by the root, add to the next coefficient, repeat.
Example 2: - Setup: $[7 | 2, -16, 19]$ - Process: Similar steps as above.
Example 3: - Setup: $[2 | 1, 5, -7, 2]$ - Process: Similar steps as above.
Example 4: - Setup: $[-4 | 1, 9, 13, -33, -16]$ - Process: Similar steps as above.
Example 5: - Setup: $[-5 | 3, -17, -32]$ - Process: Similar steps as above.
Example 6: - Setup: $[8 | 8, 64, 0, 3, -28]$ - Process: Similar steps as above.

STEP 5

Write the resulting quotient and remainder for each example.
Example 1: - Quotient: $x - 5$ - Remainder: $-3$
Example 2: - Quotient: $2n - 2$ - Remainder: $5$
Example 3: - Quotient: $y^3 + 7y^2 + 7y + 7$ - Remainder: $16$
Example 4: - Quotient: $c^3 + 5c^2 - 7c - 5$ - Remainder: $4$
Example 5: - Quotient: $3k - 32$ - Remainder: $-192$
Example 6: - Quotient: $8x^3 + 128x^2 + 1027x + 8219$ - Remainder: $65624$

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