If f(1) = 0, what are all the roots of the function f(x) = x³ + 3x² - x - 3. Use the remainder theorem.
x = -1, x = 1, or x = 3
x = -3, x = -1, or x = 1
x = -3 or x = 1
x = -1 or x = 3

Solution:

Given f(1) = 0, clearly (x - 1) is one of the factors of f(x) = x³ + 3x² - x - 3

Using remainders theorem,

f(x) = q(x)(x - 1) = r(x)

Since (x - 1) is a factor, r(x) = 0. Now to find the remaining factor i.e.,

p(x) = f(x)/(x + 1)

Using long division method,

The quotient is x² + 4x + 3.

To find remaining factors,

x² + 4x + 3 = 0

x² + 3x + x + 3 = 0

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

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

(x + 1) = 0 ⇒ x = -1

(x + 3) = 0 ⇒ x = -3

Therefore, the roots of the given function are -3, -1, 1.

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If f(1) = 0, what are all the roots of the function f(x) = x³ + 3x² - x - 3. Use the remainder theorem.

Summary:

The roots of the given function f(x) = x³ + 3x² - x - 3 are -3, -1, 1.