Answer :
- Divide $3x^4$ by $3x$ to get $x^3$.
- Multiply $(3x+2)$ by $x^3$ to get $3x^4 + 2x^3$.
- Subtract $2x^3$ from $23x^3$ to get $21x^3$.
- Divide $21x^3$ by $3x$ to get $7x^2$.
- Multiply $(3x+2)$ by $7x^2$ to get $21x^3 + 14x^2$.
- Subtract $14x^2$ from $14x^2$ to get $0$.
- Divide $-3x$ by $3x$ to get $-1$.
- Multiply $(3x+2)$ by $-1$ to get $-3x - 2$.
- The quotient is $\boxed{x^3 + 7x^2 - 1}$.
### Explanation
1. Understanding the Problem
We are given the dividend $3x^4 + 23x^3 + 14x^2 - 3x - 2$ and the divisor $3x + 2$. We need to find the quotient using the box method. The box method is a visual representation of polynomial long division.
2. Finding the First Term of the Quotient
First, we set up the box method. We divide the first term of the dividend, $3x^4$, by the first term of the divisor, $3x$, to get $x^3$. This is the first term of the quotient. We multiply the divisor $(3x + 2)$ by $x^3$ to get $3x^4 + 2x^3$. We place $3x^4$ in the first box.
3. Finding the Second Term of the Quotient
Next, we subtract $2x^3$ from $23x^3$ to get $21x^3$. We divide $21x^3$ by $3x$ to get $7x^2$. This is the second term of the quotient. We multiply the divisor $(3x + 2)$ by $7x^2$ to get $21x^3 + 14x^2$.
4. Finding the Third Term of the Quotient
Next, we subtract $14x^2$ from $14x^2$ to get $0$. We divide $0$ by $3x$ to get $0$. We bring down the next term, $-3x$. We divide $-3x$ by $3x$ to get $-1$. This is the third term of the quotient. We multiply the divisor $(3x + 2)$ by $-1$ to get $-3x - 2$.
5. Finding the Remainder and the Quotient
Finally, we subtract $-3x$ from $-3x$ to get $0$, and we subtract $-2$ from $-2$ to get $0$. The remainder is $0$. The quotient is $x^3 + 7x^2 - 1$.
6. Final Answer
The quotient when $3x^4 + 23x^3 + 14x^2 - 3x - 2$ is divided by $3x + 2$ is $x^3 + 7x^2 - 1$.
### Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design, signal processing, and cryptography. For example, in control systems, polynomial division can be used to simplify transfer functions, which describe the relationship between the input and output of a system. By dividing polynomials, engineers can reduce the complexity of the transfer function and make it easier to analyze and design the control system.
- Multiply $(3x+2)$ by $x^3$ to get $3x^4 + 2x^3$.
- Subtract $2x^3$ from $23x^3$ to get $21x^3$.
- Divide $21x^3$ by $3x$ to get $7x^2$.
- Multiply $(3x+2)$ by $7x^2$ to get $21x^3 + 14x^2$.
- Subtract $14x^2$ from $14x^2$ to get $0$.
- Divide $-3x$ by $3x$ to get $-1$.
- Multiply $(3x+2)$ by $-1$ to get $-3x - 2$.
- The quotient is $\boxed{x^3 + 7x^2 - 1}$.
### Explanation
1. Understanding the Problem
We are given the dividend $3x^4 + 23x^3 + 14x^2 - 3x - 2$ and the divisor $3x + 2$. We need to find the quotient using the box method. The box method is a visual representation of polynomial long division.
2. Finding the First Term of the Quotient
First, we set up the box method. We divide the first term of the dividend, $3x^4$, by the first term of the divisor, $3x$, to get $x^3$. This is the first term of the quotient. We multiply the divisor $(3x + 2)$ by $x^3$ to get $3x^4 + 2x^3$. We place $3x^4$ in the first box.
3. Finding the Second Term of the Quotient
Next, we subtract $2x^3$ from $23x^3$ to get $21x^3$. We divide $21x^3$ by $3x$ to get $7x^2$. This is the second term of the quotient. We multiply the divisor $(3x + 2)$ by $7x^2$ to get $21x^3 + 14x^2$.
4. Finding the Third Term of the Quotient
Next, we subtract $14x^2$ from $14x^2$ to get $0$. We divide $0$ by $3x$ to get $0$. We bring down the next term, $-3x$. We divide $-3x$ by $3x$ to get $-1$. This is the third term of the quotient. We multiply the divisor $(3x + 2)$ by $-1$ to get $-3x - 2$.
5. Finding the Remainder and the Quotient
Finally, we subtract $-3x$ from $-3x$ to get $0$, and we subtract $-2$ from $-2$ to get $0$. The remainder is $0$. The quotient is $x^3 + 7x^2 - 1$.
6. Final Answer
The quotient when $3x^4 + 23x^3 + 14x^2 - 3x - 2$ is divided by $3x + 2$ is $x^3 + 7x^2 - 1$.
### Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design, signal processing, and cryptography. For example, in control systems, polynomial division can be used to simplify transfer functions, which describe the relationship between the input and output of a system. By dividing polynomials, engineers can reduce the complexity of the transfer function and make it easier to analyze and design the control system.
