Use synthetic division to find the quotient and remainder when 7 x Superscript 6 Baseline minus 2 x Superscript 4 Baseline plus x squared plus 5 is divided by x minus 2.

To use synthetic division to find the quotient and remainder when dividing \(7x^6 - 2x^4 + x^2 + 5\) by \(x - 2\), follow these steps:

1. **Set up the synthetic division.**  
The divisor \(x - 2\) gives a root of \(2\). We'll use this root for synthetic division.

2. **Write the coefficients of the dividend.**  
Include coefficients for all powers of \(x\), including those with zero coefficients.
The polynomial is \(7x^6 - 2x^4 + x^2 + 5\), so the coefficients are:
\[ 7, 0, -2, 0, 1, 0, 5 \]

3. **Perform the synthetic division:**

\[
\begin{array}{r|rrrrrrr}
2 & 7 & 0 & -2 & 0 & 1 & 0 & 5 \\
  &   & 14 & 28 & 52 & 104 & 210 & 420 \\
\hline
  & 7 & 14 & 26 & 52 & 105 & 210 & 425 \\
\end{array}
\]

Here's a step-by-step explanation of each row in the table:

- **First row:** The root from the divisor and the coefficients from the dividend polynomial.
- **Second row:** Multiply the last entry of the result row by the root and add to the next coefficient.

\[
\begin{array}{r|rrrrrrr}
2 & 7 & 0 & -2 & 0 & 1 & 0 & 5 \\
  &   & 14 & 28 & 52 & 104 & 210 & 420 \\
\hline
  & 7 & 14 & 26 & 52 & 105 & 210 & 425 \\
\end{array}
\]

### Steps:
1. Bring down the first coefficient, \(7\).
2. Multiply \(7\) by \(2\) (the root from the divisor), giving \(14\).
3. Add \(14\) to the next coefficient \(0\), giving \(14\).
4. Multiply \(14\) by \(2\), giving \(28\).
5. Add \(28\) to the next coefficient \(-2\), giving \(26\).
6. Multiply \(26\) by \(2\), giving \(52\).
7. Add \(52\) to the next coefficient \(0\), giving \(52\).
8. Multiply \(52\) by \(2\), giving \(104\).
9. Add \(104\) to the next coefficient \(1\), giving \(105\).
10. Multiply \(105\) by \(2\), giving \(210\).
11. Add \(210\) to the next coefficient \(0\), giving \(210\).
12. Multiply \(210\) by \(2\), giving \(420\).
13. Add \(420\) to the next coefficient \(5\), giving \(425\).

### Result:
The bottom row (excluding the last value) gives the coefficients of the quotient polynomial, and the last value is the remainder.

Thus, the quotient is:
\[7x^5 + 14x^4 + 26x^3 + 52x^2 + 105x + 210\]

And the remainder is:
\[425\]

### Final Answer:
The quotient is \(7x^5 + 14x^4 + 26x^3 + 52x^2 + 105x + 210\) and the remainder is \(425\).

Would you like further details or have any questions?

Here are 5 more questions you might want to ask:
1. Can you explain the process of long division for polynomials?
2. How do you find the roots of a polynomial?
3. What are the applications of synthetic division in real life?
4. How do you factorize a polynomial?
5. Can you solve a polynomial equation using synthetic division?

**Tip:** When using synthetic division, always ensure you account for all terms in the polynomial, including those with a coefficient of zero.

Learn how to use synthetic division to find the quotient and remainder when dividing 7x^6 - 2x^4 + x^2 + 5 by x - 2. This problem involves polynomial division using synthetic division method, suitable for high school level students.

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