What is the completely factored form of 3x⁵ - 7x⁴ + 6x² - 14x?

Solution:

Given polynomial f(x) = 3x⁵ - 7x⁴ + 6x² - 14x

3x⁵ - 7x⁴ + 6x² - 14x = 0

x(3x⁴ - 7x³ + 6x - 14) = 0

x[x³(3x - 7) + 2(3x - 7)] = 0

x(x³ + 2)(3x - 7) = 0

Example:

Find the factors of P(x) = 6x³ - 13x² + 4x + 3

Solution:

Put x = 1 in the P(x) = 6(1)³ - 13(1)² + 4(1) + 3 = 0 Now divide P(x) using (x -1) by synthetic division

Thus P(x) = (x - 1)(6x² - 7x - 3)

Now consider (6x² - 7x - 3) = 0

⇒ 6x² -9x + 2x - 3 = 0

⇒ 3x(2x - 3) + 1(2x - 3) = 0

⇒ (2x - 3)(3x + 1) = 0

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What is the completely factored form of 3x⁵ - 7x⁴ + 6x² - 14x?

Summary:

The completely factored form of 3x⁵ - 7x⁴ + 6x² - 14x is x(x³ + 2)(3x - 7) = 0.