if two zeros of the polynomial x power 4 - 6 x cube - 26 X square + 138 x-35 are 2 +or -root 3 find the other zeros
Answers
Polynomials
Solution.
The given polynomial is
p(x) = x⁴ - 6x³ - 26x² + 138x - 35
∵ (2 + √3) and (2 - √3) are the zeroes of p(x),
x - (2 + √3), x - (2 - √3) are factors of p(x)
⇒ {x - (2 + √3)} {x - (2 - √3)} is a factor of p(x)
⇒ x² - (2 - √3 + 2 + √3) x + (2 + √3) (2 - √3) is a factor of p(x)
⇒ x² - 4x + (4 - 3) is a factor of p(x)
⇒ x² - 4x + 1 is a factor of p(x)
∴ x⁴ - 6x³ - 26x² + 138x - 35
= x⁴ - 4x³ + x² - 2x³ + 8x² - 2x - 35x² + 140x - 35
= x² (x² - 4x + 1) - 2x (x² - 4x + 1) - 35 (x² - 4x + 1)
= (x² - 4x + 1) (x^2 - 2x - 35)
∴ another factor of p(x) is (x² - 2x - 35).
∴ x² - 2x - 35
= x² - 7x + 5x - 35
= x (x - 7) + 5 (x - 7)
= (x - 7) (x + 5)
∴ (x - 7), (x + 5) are two factors of p(x)
∴ the other two zeroes are 7, - 5.
Read more on Brainly.in
If two zeros of a polynomial p(x) = x⁴ - 5x³ - x² + 35x - 42 are 2 and 3, find other two zeros of the polynomial.
- https://brainly.in/question/17307670
Answer:
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