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dusya [7]
3 years ago
9

Complete the steps to solve the polynomial equation x3 – 21x = –20. According to the rational root theorem, which number is a po

tential root of the polynomial?
Mathematics
1 answer:
jeka943 years ago
3 0

Answer:

Zeroes : 1, 4 and -5.

Potential roots: \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20.

Step-by-step explanation:

The given equation is

x^3-21x=-20

It can be written as

x^3+0x^2-21x+20=0

Splitting the middle terms, we get

x^3-x^2+x^2-x-20x+20=0

x^2(x-1)+x(x-1)-20(x-1)=0

(x-1)(x^2+x-20)=0

Splitting the middle terms, we get

(x-1)(x^2+5x-4x-20)=0

(x-1)(x(x+5)-4(x+5))=0

(x-1)(x+5)(x-4)=0

Using zero product property, we get

x-1=0\Rightarrow x=1

x-4=0\Rightarrow x=4

x+5=0\Rightarrow x=-5

Therefore, the zeroes of the equation are 1, 4 and -5.

According to rational root theorem, the potential root of the polynomial are

x=\dfrac{\text{Factor of constant}}{\text{Factor of leading coefficient}}

Constant = 20

Factors of constant ±1, ±2, ±4, ±5, ±10, ±20.

Leading coefficient= 1

Factors of leading coefficient ±1.

Therefore, the potential root of the polynomial are \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20.

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