What divisor is represented by the synthetic division below?
$-\begin{array}{c}-5 \left\lvert\, \begin{array}{cccr}2 & 10 & 1 & 5 \\ & -10 & 0 & -5 \\ 2 & 0 & 1 & 0\end{array}\right.\end{array}$
Answer (1)
The number outside the synthetic division, − 5 , represents the value of c in the divisor x − c .
Substitute c = − 5 into the expression x − c .
Simplify x − ( − 5 ) to find the divisor.
The divisor is x + 5 .
Explanation
Understanding Synthetic Division The synthetic division is set up to find the quotient and remainder when a polynomial is divided by a linear expression of the form x − c . The number outside the division, in this case, − 5 , represents the value of c .
Determining the Divisor The divisor in synthetic division is of the form x − c , where c is the number outside the division. In this case, c = − 5 . Therefore, the divisor is x − ( − 5 ) = x + 5 .
Final Answer Thus, the divisor represented by the synthetic division is x + 5 .
Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − c . It's used in various applications, such as finding roots of polynomials, simplifying expressions, and solving algebraic equations. For example, if you want to determine if ( x − 2 ) is a factor of the polynomial x 3 − 4 x 2 + 5 x − 2 , you can use synthetic division with c = 2 . If the remainder is 0, then ( x − 2 ) is a factor. This technique is also used in engineering to analyze systems and in computer science for algorithm optimization.
