Use synthetic division to find the quotient and remainder.
[tex]$\frac{x^4-25}{x+2}$[/tex]
quotient [$\square$]
remainder [$\square$]
Answer (2)
The quotient of x 4 − 25 divided by x + 2 is x 3 − 2 x 2 + 4 x − 8 and the remainder is − 9 .
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Set up the synthetic division with the coefficients of the dividend (1, 0, 0, 0, -25) and the value from the divisor (-2).
Perform synthetic division.
Read the coefficients of the quotient from the result of the synthetic division: 1, -2, 4, -8.
Identify the remainder from the result of the synthetic division: -9.
The quotient is x 3 − 2 x 2 + 4 x − 8 and the remainder is − 9 .
Explanation
Understanding the Problem We want to divide the polynomial x 4 − 25 by x + 2 using synthetic division. First, we rewrite the dividend as x 4 + 0 x 3 + 0 x 2 + 0 x − 25 to include all powers of x down to the constant term. The divisor is x + 2 , so we use − 2 in the synthetic division.
Setting up Synthetic Division Now, we set up the synthetic division table. The coefficients of the dividend are 1, 0, 0, 0, and -25. We write these coefficients in a row, and we write -2 to the left.
Performing Synthetic Division We perform the synthetic division as follows:
Bring down the first coefficient (1).
Multiply -2 by 1 to get -2, and write it below the second coefficient (0).
Add 0 and -2 to get -2.
Multiply -2 by -2 to get 4, and write it below the third coefficient (0).
Add 0 and 4 to get 4.
Multiply -2 by 4 to get -8, and write it below the fourth coefficient (0).
Add 0 and -8 to get -8.
Multiply -2 by -8 to get 16, and write it below the last coefficient (-25).
Add -25 and 16 to get -9.
The last number, -9, is the remainder. The other numbers (1, -2, 4, -8) are the coefficients of the quotient.
Determining Quotient and Remainder The quotient is 1 x 3 − 2 x 2 + 4 x − 8 , and the remainder is -9.
Final Answer Therefore, when x 4 − 25 is divided by x + 2 , the quotient is x 3 − 2 x 2 + 4 x − 8 and the remainder is -9.
Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − a . It's often used in algebra to find the roots of polynomials or to simplify rational expressions. For example, if you're designing a bridge and need to calculate the bending moment of a beam, you might encounter polynomial equations. Synthetic division can help you quickly find the roots of these equations, which are crucial for determining the beam's stability and safety. It provides an efficient way to handle polynomial division in various engineering and scientific applications.
