18 Consider the following statements
(i) x – 2 is a factor of x3 – 3x2 + 4x – 4.
(ii) x + 1 is a factor of 2x3 + 4x + 6.
(iii) x – 1 is a factor of x5 + x4 – x3 + x2 – x + 1
In these statements
(a) 1 and 2 are correct
(b) 1, 2 and 3 are correct
(c) 2 and 3 are correct
(d) 1 and 3 are correct
Answers
Answer:
a) 1 and 2 are correct
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Answer:
Hence,The correct option is (a) 1 and 2 are correct.
Step-by-step explanation:
i) If (x - 2) is the factor of the given polynomial, then we must get the value of the polynomial as 0, if we substitute 2 in place of x. [Here - becomes +]
x³ - 3x² + 4x - 4 = 2³ - 3(2)² + 4 × 2 - 4
2³ - 3(2)² + 4 × 2 - 4 = 8 - 12 + 8 - 4 = 0
Hence, (x - 2) is the factor of x³ - 3x² + 4x - 4.
Therefore, it is correct.
ii) If (x + 1) is the factor of the given polynomial, then we must get the value of the polynomial as 0, if we substitute -1 in place of x. [Here + becomes -]
2x³ + 4x + 6 = 2(-1)³ + 4 × -1 + 6
2(-1)³ + 4 × -1 + 6 = -2 + (-4) + 6 = -2 -4 + 6 = -6 + 6 = 0
Hence, (x + 1) is the factor of 2x³ + 4x + 6.
Therefore, it is correct.
iii) If (x - 1) is the factor of the given polynomial, then we must get the value of the polynomial as 0, if we substitute 1 in place of x. [Here - becomes +]
x⁵ + x⁴ - x³ + x² - x + 1 = (1)⁵ + (1)⁴ - (1)³ + (1)² - 1 + 1
(1)⁵ + (1)⁴ - (1)³ + (1)² - 1 + 1 = 1 + 1 - 1 + 1 - 1 + 1 = 2
2 ≠ 0
Hence, (x - 2) is not the factor of x⁵ + x⁴ - x³ + x² - x + 1
Therefore, it is wrong.
Hence,The correct option is (a) 1 and 2 are correct.
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