Find the quotient and remainder using synthetic division: 
x
3
−
8
x
2
+
70
x
−
5



To use synthetic division, we need to divide the given polynomial by a binomial of the form \( x - c \). However, in your question, the divisor is not mentioned. Assuming the divisor is of the form \( x - c \), I will walk you through how to apply synthetic division to a polynomial \( x^3 - 8x^2 + 70x - 5 \).

If the divisor is \( x - 1 \), we will follow these steps:

### Step 1: Set up synthetic division
We write the coefficients of the polynomial \( x^3 - 8x^2 + 70x - 5 \), which are: 
\[ [1, -8, 70, -5] \]

The divisor \( x - 1 \) means \( c = 1 \).

### Step 2: Perform synthetic division
1. Bring down the first coefficient (which is \( 1 \)) to the bottom row.
2. Multiply \( c = 1 \) by the number in the bottom row (currently \( 1 \)), and write the result under the next coefficient.
3. Add this result to the next coefficient and write the sum below the line.
4. Repeat the process.

Let's perform the division:

\[
\begin{array}{r|rrrr}
  1 & 1  & -8 & 70 & -5 \\
    &    & 1  & -7 & 63 \\
\hline
    & 1  & -7 & 63 & 58 \\
\end{array}
\]

### Step 3: Interpret the result
The bottom row \( [1, -7, 63] \) represents the quotient, and the last number \( 58 \) is the remainder.

So, the quotient is:
\[
x^2 - 7x + 63
\]
and the remainder is:
\[
58
\]

Thus, the quotient is \( x^2 - 7x + 63 \) and the remainder is \( 58 \).

---

Do you need more details or have any questions?

Here are 5 related questions:
1. What if the divisor was \( x + 2 \)? How would the synthetic division change?
2. How is synthetic division different from polynomial long division?
3. Can synthetic division be used for non-linear divisors?
4. How can you check if the remainder is correct?
5. Why is synthetic division more efficient than long division?

**Tip:** In synthetic division, the sign of the divisor \( x - c \) matters. Always use the opposite of the binomial term when setting up your division!

Find the quotient and remainder using synthetic division: x^3 − 8x^2 + 70x − 5

Learn how to find the quotient and remainder using synthetic division for the polynomial x^3 − 8x^2 + 70x − 5. Follow a detailed, step-by-step guide suitable for students in Grades 9-11. This example explains the synthetic division method with clear instructions and insights into algebraic concepts.

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