Divide using synthetic division. Begin the process as shown.
[tex]\left(3 x^2+x-14\right) \div(x-2) \quad 2 \left\lvert\,\begin{array}{lll}
& 3 & 1
\end{array}-14\right.[/tex]
[tex]\left(3 x^2+x-14\right) \div(x-2)=[/tex]
[tex]\square[/tex]
(Simplify your answer.)
Answer (2)
Set up the synthetic division table with the divisor (2) and the coefficients of the dividend (3, 1, -14).
Bring down the first coefficient (3).
Multiply the divisor (2) by the brought-down coefficient (3) and add the result to the next coefficient (1), obtaining 7.
Multiply the divisor (2) by 7 and add the result to the last coefficient (-14), obtaining 0.
The quotient is 3 x + 7 and the remainder is 0, so the final answer is 3 x + 7 .
Explanation
Understanding the problem We are asked to divide the polynomial 3 x 2 + x − 14 by x − 2 using synthetic division. Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − a . In this case, a = 2 .
Setting up the synthetic division We set up the synthetic division as follows:
2 | 3 1 -14
|
|__________
First, we bring down the leading coefficient, which is 3:
2 | 3 1 -14
|
|__________
3
First multiplication Next, we multiply the 2 by the 3 and write the result (6) under the next coefficient (1):
2 | 3 1 -14
| 6
|__________
3
Adding the coefficients Now, we add 1 and 6 to get 7:
2 | 3 1 -14
| 6
|__________
3 7
Second multiplication We multiply 2 by 7 to get 14 and write it under -14:
2 | 3 1 -14
| 6 14
|__________
3 7
Final result Finally, we add -14 and 14 to get 0:
2 | 3 1 -14
| 6 14
|__________
3 7 0
The numbers 3 and 7 are the coefficients of the quotient, and 0 is the remainder. Therefore, the quotient is 3 x + 7 and the remainder is 0.
Conclusion Thus, x − 2 3 x 2 + x − 14 = 3 x + 7 .
Examples
Synthetic division is a streamlined way to divide polynomials, useful in various applications. For instance, when designing a bridge, engineers use polynomial functions to model the load distribution. If they need to determine how the load behaves under specific conditions, they might divide the polynomial by a factor representing those conditions. The result helps them understand the remaining load distribution, ensuring the bridge's structural integrity and safety. This method simplifies complex calculations, making it easier to analyze and design safe structures.
Using synthetic division on 3 x 2 + x − 14 by x − 2 , we find the quotient to be 3 x + 7 with a remainder of 0 . Thus, the result of the division is 3 x + 7 .
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