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Chapter 2: Problem 78

Use synthetic division to show that 5 is a solution of $$x^{4}-4 x^{3}-9 x^{2}+16 x+20=0$$ Then solve the polynomial equation.

Short Answer

Expert verified
5 is confirmed to be a root of the polynomial after performing synthetic division. Following the division, we obtained a cubic equation \(x^{3}+x^{2}-4x-4 = 0\) which has 1 as a root. Using synthetic division or factoring methods, we could find the remaining roots of this cubic and hence the polynomial equation.

Step by step solution

01

Verify root using synthetic division

To verify if 5 is a root of the given polynomial, we need to perform synthetic division. For synthetic division, write the coefficients of the polynomial in a row and place 5 outside the row. Perform synthetic division and if the last element in the result is zero, it means 5 is a root of the polynomial. The coefficients of the polynomial are 1,-4,-9,16, and 20.
02

Perform Synthetic Division

First, copy the first coefficient to the bottom row. Then, multiply the number you just wrote by 5 and write the result under the second coefficient. Add the two numbers in the second column, write the result in the bottom row, and repeat the process. In this case, with 5 as the root: \[\begin{array}{c|ccccc} 5 & 1 & -4 & -9 & 16 & 20 \ \hline & 5 & 5 & -20 & -20 & 0 \end{array}\] The final row of coefficients after synthetic division is \(1, 1, -4, -4, 0\). Since the remainder (last number in the bottom row) is zero, 5 is indeed a root of the polynomial.
03

Factorize the Polynomial

Now using the bottom row (except the remainder) from the synthetic division as coefficients of a new polynomial: \(x^{3}+x^{2}-4x-4 = 0 \). To solve this cubic equation, use factoring or the cubic formula.
04

Solving the Cubic

The cubic formula or factoring might be difficult to use in this case. Thus, using rational root theorem or making an educated guess about the potential roots and verifying them by synthetic division/ substitution seems more practical. In this case, 1 is a root of this cubic equation. To find the other roots, you could again use synthetic division with 1 as a root, or solve the quadratic obtained after factoring out \(x-1\) from the cubic expression.

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