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Chapter 2: Problem 60

Use synthetic division to show that \(x\) is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. $$x^{3}-28 x-48=0, \quad x=-4$$

Short Answer

Expert verified
By the process of synthetic division, we confirm that \(x = -4\) is a solution of the polynomial \(x^3 - 28x - 48 = 0\). After the division, the factored form of the polynomial results in \((x + 4)(...)\). Solving the remaining part of the polynomial yields the remaining real solutions.

Step by step solution

01

Setup for Synthetic Division

Start synthetic division by setting up the coefficients of the cubic polynomial and the proposed solution \(x=-4\). The coefficients are 1 (for \(x^3\)), 0 (for the missing \(x^2\) term), -28 (for \(x\)) and -48 (for the constant term).
02

Perform Synthetic Division

To perform synthetic division, start by bringing down the first coefficient (1 in this case) to the bottom row. Multiply this number by the proposed solution (-4), and write the result under the next coefficient (0). Add these two together to get the next number in the bottom row, and repeat for the remaining coefficients.
03

Check if x=-4 is a Solution

After the division, the last number in the bottom row is the remainder. If remainder is zero, \(x=-4\) is indeed a solution of the polynomial equation. The remaining numbers in the bottom row represent the coefficients of the quotient polynomial.
04

Write Down the Factored Form of Polynomial

Write the factored form of the polynomial by taking the result of synthetic division, which is a quadratic polynomial, and factorize it, if possible. In this case, the polynomial will be in the form \((x + 4)(...)\), where '...' stands for the factored form of the resultant quadratic polynomial from step 3.
05

Solve for x in the Factored Polynomial

Now, solve for the values of \(x\) in the factored form of the polynomial. These are the real solutions for the original equation.

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Most popular questions from this chapter

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$2 x^{2}+b x+5=0$$

Think About It \(\quad\) A cubic polynomial function \(f\) has real zeros \(-2, \frac{1}{2},\) and \(3,\) and its leading coefficient is negative. Write an equation for \(f\) and sketch its graph. How many different polynomial functions are possible for \(f ?\)

Find all real zeros of the function. $$f(z)=12 z^{3}-4 z^{2}-27 z+9$$

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x+4=0$$

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-2 x$$
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