If F(1) = 0, What Are All The Roots Of The Function F(X) = X^3 + 3x^2 - X - 3? Use The Remainder Theorem., A) X = 20131, X = 1, Or X = 3, B) X = 20133

If f(1) = 0, what are all the roots of the function f(x) = x^3 + 3x^2 - x - 3? Use the Remainder Theorem.

A) x = –1, x = 1, or x = 3
B) x = –3, x = –1, or x = 1
C) x = –3 or x = 1
D) x = –1 or x = 3

 

Answer:

Therefore the roots of x³ + 3x² - x - 3 are:

(B) x = -3; x = -1 and x = 1.

Step-by-step explanation:

If f(1) = 0, therefore there is no remainder if the given polynomial is divided by x-1 and 1 is a root of the polynomial.

x - 1 = 0

x = 1

Divide the polynomial by (x-1) using synthetic division:

(x³ + 3x² = x - 3) ÷ (x - 1)  ⇒ x=1

The coefficients are:

1 + 3  - 1 - 3

After bringing down the first coefficient as the coefficient of first term of the quotient, follow the steps:

a) Bring down 1 ⇒ 1

b) (1 × 1) + 3  ⇒ 4

c) (4 × 1) - 1 = 3

d) (3 × 1)  - 3 = 0

The quotient is 1x² + 4x + 3  or x² + 4x + 3.

Factor x² + 4x + 3 to find the remaining roots:

(x + 3) (x + 1) = 0

x+3 = 0

x = -3

x + 1 = 0

x = -1

Therefore the roots of x³ + 3x² - x - 3 are (B) x = -3; x = -1 and x = 1.