Question

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Answer 1

Answer:

2x³ - 11x² + 17x - 6 = (x - 1)(x - 3)(2x - 1)

Step-by-step explanation:

Factor theorem is done using trial and error method

Let p(x) = 2x³ - 11x² + 17x - 6

Let (x - 1) be a factor of p(x)

then,

x - 1 = 0

x = 1

so,

p(1) = 2(1)³ - 11(1)² + 17(1) - 6

= 2 - 11 + 17 - 6

= 2

so, (x - 1) is not a factor since it gave a remainder of 2

Now, let (x + 1) be a factor of p(x)

then,

x + 1 = 0

x = -1

p(-1) = 2(-1)³ - 11(-1)² + 17(-1) - 6

= 2(-1) - 11(1) - 17 - 6

= -2 - 11 - 17 - 6

= -36

So, (x + 1) is also not a factor of p(x)

Now,

Let (x - 2) be a factor of p(x)

x - 2 = 0

x = 2

p(2) = 2(2)³ - 11(2)² + 17(2) - 6

= 2(8) - 11(4) + 34 - 6

= 16 - 44 + 34 - 6

= 0

Thus, (x - 2) is a factor of p(x)

Now, let (x - 2) = g(x)

Thus, to get the other factor we must divide p(x) by g(x)

2x² - 7x + 3

——————————

x - 2 | 2x³ - 11x² + 17x - 6

– (2x³ - 4x²)

——————

0 -7x² + 17x

– (-7x² + 14x)

———————

0 + 3x - 6

– (3x - 6)

—————

0

—————

Thus,

2x³ - 11x² + 17x - 6 = (x - 2)(2x² - 7x + 3)

p(x) = (x - 2)(2x² - 7x + 3)

Now, we must again factorise 2x² - 7x + 3

so,

2x² - 7x + 3

ax² + bx + c

where a = 2, b = -7, c = 3

By splitting the middle term method,

Sum = b = -7

Product = a × c = 6

So, Factors are -1 and - 6

2x² - 1x - 6x + 3= 0

x(2x - 1) - 3(2x - 1) = 0

(x - 3)(2x - 1) = 0

Thus,

the factors of p(x) are (x - 1), (x - 3) and (2x - 1)

So,

2x³ - 11x² + 17x - 6 = (x - 1)(x - 3)(2x - 1)

Hope it helped and you understood it........All the best