Explain the mistake in the synthetic division below. ## Synthetic Division Error Analysis### Problem StatementWhat is the mistake in the synthetic division process shown below?### Division ExpressionThe division expression is:\[\frac{3m^3 - 2m^2 - 150}{m - 4}\]### Synthetic Division Process1. **Setup:** - Divisor: \( m - 4 \), so the root used is \( 4 \). - Coefficients of the polynomial: \( 3, -2, -150 \).2. **Execution:** - Write the divisor root: \( 4 \) on the left. - Write the polynomial coefficients: \( 3, -2, -150 \).3. **Process:** - Bring down the first coefficient: \( 3 \). - Multiply \( 3 \) by \( 4 \): add \( 12 \) under the next coefficient. - Add: \(-2 + 12 = 10\). - Multiply \( 10 \) by \( 4 \): add \( 40 \) under the next coefficient. - Add: \(-150 + 40 = -110\).### Explanation of ErrorsThe synthetic division should start with the coefficients of the polynomial including the constant term. Also, since the polynomial is of degree 3, it should have coefficients for each power of \( m \) starting from the highest to zero, meaning the setup might be missing necessary zero coefficients. In this case, the polynomial terms might have been inaccurately represented or reduced prematurely. Additionally, each step in the synthetic division needs careful verification, especially the carrying down and multiplying processes. **Long Division of Polynomials**Consider the polynomial division:Dividend: \( 2x^3 - 3x^2 + 4x + 36 \) Divisor: \( x + 2 \)**Step 1: Determine the Zero of the Divisor**Set the divisor equal to zero:\[ x + 2 = 0 \] \[ x = -2 \]**Step 2: Synthetic Division Setup**Using synthetic division, the process involves:- Coefficients of the dividend: 2, -3, 4, 36- Zero of the divisor: -2**Step 3: Synthetic Division Process**- Write down the leading coefficient (2). Multiply it by -2 to get -4. - Add -3 and -4 to get -7. - Multiply -7 by -2 to get 14. - Add 4 and 14 to get 18. - Multiply 18 by -2 to get -36. - Add 36 and -36 to get 0.The synthetic division is shown as:```-2 | 2 -3 4 36 | -4 -14 -36 ----------------- 2 -7 18 0```- Quotient: \( 2x^2 - 7x + 18 \)- Remainder: 0**Conclusion**Since the remainder is 0, \( x + 2 \) and the quotient \( 2x^2 - 7x + 18 \) are factors of the original polynomial \( 2x^3 - 3x^2 + 4x + 36 \).**Verification Equation**\[ (x + 2)(2x^2 - 7x + 18) = 2x^3 - 3x^2 + 4x + 36 \]This confirms that the division was performed correctly.

We need to find error in division.

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What is the mistake for the synthetic division below? 나 3 J -2 a 12 -150 40 3m3-2m2-150 m-4

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Explain the mistake in the synthetic division below.

## Synthetic Division Error Analysis

### Problem Statement
What is the mistake in the synthetic division process shown below?

### Division Expression
The division expression is:
\[
\frac{3m^3 - 2m^2 - 150}{m - 4}
\]

### Synthetic Division Process

1. **Setup:**

- Divisor: \( m - 4 \), so the root used is \( 4 \).
- Coefficients of the polynomial: \( 3, -2, -150 \).

2. **Execution:**

- Write the divisor root: \( 4 \) on the left.
- Write the polynomial coefficients: \( 3, -2, -150 \).

3. **Process:**

- Bring down the first coefficient: \( 3 \).
- Multiply \( 3 \) by \( 4 \): add \( 12 \) under the next coefficient.
- Add: \(-2 + 12 = 10\).
- Multiply \( 10 \) by \( 4 \): add \( 40 \) under the next coefficient.
- Add: \(-150 + 40 = -110\).

### Explanation of Errors
The synthetic division should start with the coefficients of the polynomial including the constant term. Also, since the polynomial is of degree 3, it should have coefficients for each power of \( m \) starting from the highest to zero, meaning the setup might be missing necessary zero coefficients. In this case, the polynomial terms might have been inaccurately represented or reduced prematurely. Additionally, each step in the synthetic division needs careful verification, especially the carrying down and multiplying processes.

**Long Division of Polynomials**

Consider the polynomial division:

Dividend: \( 2x^3 - 3x^2 + 4x + 36 \)
Divisor: \( x + 2 \)

**Step 1: Determine the Zero of the Divisor**

Set the divisor equal to zero:

\[ x + 2 = 0 \]
\[ x = -2 \]

**Step 2: Synthetic Division Setup**

Using synthetic division, the process involves:

- Coefficients of the dividend: 2, -3, 4, 36
- Zero of the divisor: -2

**Step 3: Synthetic Division Process**

- Write down the leading coefficient (2). Multiply it by -2 to get -4.
- Add -3 and -4 to get -7.
- Multiply -7 by -2 to get 14.
- Add 4 and 14 to get 18.
- Multiply 18 by -2 to get -36.
- Add 36 and -36 to get 0.

The synthetic division is shown as:

```
-2 | 2 -3 4 36
| -4 -14 -36
-----------------
2 -7 18 0
```

- Quotient: \( 2x^2 - 7x + 18 \)
- Remainder: 0

**Conclusion**

Since the remainder is 0, \( x + 2 \) and the quotient \( 2x^2 - 7x + 18 \) are factors of the original polynomial \( 2x^3 - 3x^2 + 4x + 36 \).

**Verification Equation**

\[ (x + 2)(2x^2 - 7x + 18) = 2x^3 - 3x^2 + 4x + 36 \]

This confirms that the division was performed correctly.

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