Precalculus

9th Edition

ISBN: 9780321716835

Author: Michael Sullivan

Publisher: Addison Wesley

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Question

Chapter 8, Problem 10CR

To determine

The solution of the equation $3x5−10x4+21x3−42x2+36x−8=0$ in complex number system.

Expert Solution & Answer

Answer to Problem 10CR

Solution:

The solution of the equation is $x=1,  2,  13,  2i,  −2i$

Explanation of Solution

Given information:

The equation $3x5−10x4+21x3−42x2+36x−8=0$

Explanation:

Consider the equation $3x5−10x4+21x3−42x2+36x−8=0$

Let $f(x)=3x5−10x4+21x3−42x2+36x−8$

To solve this equation, use the rational root theorem to find its possible roots

Rational root theorem states that for a polynomial equation with integer coefficients

$anxn+an−1xn−1+.......+a1x1+a0=0$ , if there exists a rational solution, then the possible solution can be found by checking all the numbers $±Divisors of a0Divisors of an$

On comparing with above equation $a0=−8$ and $an=3$

$Divisors of a0$ are $±1, ±2, ±4, ±8$

$Divisors of an$ are $±1, ±3$

Thus, possible roots are $±1, 2, 4, 81, 3$

That is, $±(1,13,2,23,4,43,8,83)$

Consider $x=1$ , check whether it is root or not

Substitute $x=1$ in $3x5−10x4+21x3−42x2+36x−8$

$⇒3(1)5−10(1)4+21(1)3−42(1)2+36(1)−8$

$⇒3−10+21−42+36−8=0$

Therefore, $x=1$ is a root of $3x5−10x4+21x3−42x2+36x−8=0$

Therefore, $(x−1)$ is factor of $f(x)$

Factor out $f(x)$ by synthetic division with $x=1$ as root

Therefore,

$13−1021−4236−83−714−2883−714−2880$ .

Thus, the polynomial $f(x)$ is $(x−1)(3x4−7x3+14x2−28x+8)$

Consider $x=2$ , check whether $x=2$ is root or not of $f(x)$

Substitute $x=2$ in $3x4−7x3+14x2−28x+8$

$⇒3(2)4−7(2)3+14(2)2−28(2)+8$

$⇒3(16)−7(8)+14(4)−56+8$

$⇒48−56+56−56+8=0$

Therefore, $x=2$ is root of above polynomial

Therefore, $(x−2)$ is factor of $f(x)$

That is, $(x−2)$ is a factor of $3x4−7x3+14x2−28x+8$

By synthetic division with $x=2$ as root factor out $3x4−7x3+14x2−28x+8$

Therefore, $23−714−2886−224−83−112−40$

Thus, the term $3x4−7x3+14x2−28x+8$ is $(x−2)(3x3−x2+12x−4)$

To find the factors of $3x3−x2+12x−4$

Consider $x=13$ , check whether it is root or not of $f(x)$

It could be checked by substituting $x=13$ in $3x3−x2+12x−4$

$⇒3(13)3−(13)2+12(13)−4$

$⇒19−19+4−4=0$

Thus, $x=13$ is a root of $f(x)$

By using synthetic division with $x=13$ as a root gives

$133−112−410430120$

Thus, the term $3x3−x2+12x−4$ is written as $(x−13)(3x2+12)$

Now, to factor the polynomial $3x2+12$

Factor out $3$ in $3x2+12$ it gives $3(x2+4)$

Consider $x2+4=0$

$⇒x2=−4$

$⇒x=2i, −2i$

Therefore, the equation is $(x−1)(x−2)(3x−1)(x−2i)(x+2i)=0$

Using the Zero Factor Principle: If $ab=0$ , then $a=0$ or $b=0$ or both $a=0,b=0$ .

Thus, solution of equation is $x=1,  2,  13,  2i,  −2i$

Chapter 8, Problem 9CR

Chapter 8, Problem 11CR

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The solution of the equation 3 x 5 − 10 x 4 + 21 x 3 − 42 x 2 + 36 x − 8 = 0 in complex number system.

Solution Summary: The author explains the rational root theorem, which states that for a polynomial equation with integer coefficients, the possible solution can be found by checking all numbers.