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Chapter 3: 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
So, the roots of the given polynomial equation are \(x = 5\), and three roots found from solving the polynomial equation obtained after synthetic division.

Step by step solution

01

Verification Step

To verify that 5 is a solution to the equation, substitute \(x = 5\) into the polynomial. The equation becomes \(5^4 - 4*5^3 - 9*5^2 + 16*5 + 20\). Simplify this equation, and if the result is zero, then 5 is a root of the polynomial.
02

Synthetic Division Step

If 5 is a root, then synthetic division can be used to divide the polynomial \(x^{4}-4 x^{3}-9 x^{2}+16 x+20\) by \(x - 5\). Start by writing down the coefficients (1, -4, -9, 16 and 20) and the constant (5) in a row, leaving some spaces between them: 1 -4 -9 16 20 | 5.
03

Continue Synthetic Division

Drop down the first coefficient (1). Multiply the number just wrote down (1) by 5 and write this beneath the second coefficient (-4). Add the numbers in that column (-4 and 5) to get 1. Continue this process until all of the coefficients have been used: -4 1 -40 and 16-40 -24 | -120 and 20-120 -100.
04

Write The Polynomial

Write down the result of the synthetic division as a polynomial. The coefficients of polynomial will be 1, 1, -40, and -100. This becomes \(x^{3} + x^{2} - 40x - 100\).
05

Solve The Polynomial

This polynomial will be solved for \(x\). The polynomial equation \(x^{3} + x^{2} - 40x - 100=0\) can be solved using factorization or other methods (like the Rational Root Theorem) to find the remaining roots of the original polynomial.

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