if a and b are zeros of the polynomial x² -x-6 , then find quadratic polynomial whose zeroes are (3a-2b) and (2a+3b)
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Polynomials written in form of x² - Sx + P, represent S as sum of their roots and P as product of roots.
So, here, if a and b are roots:
→a+b = 3
→ ab 2
Thus, (a + b)² = 3² a² + b² + 2ab = 9 → a² + b² = 9 - 2(-2) = 9 + 4 = 13
Let the required polynomial is x² - px + q,
SO
p=(2a + 3b) + (3a + 2b)
p=2a + 3b + 3a + 2b = 5a + 5b
p= 5(a + b) = 5(3) = 15
Whereas,
q= (2a + 3b)(3a + 2b)
q=6(a² + b²) + 13ab
q = 6(13) + 13(-2) = 13(6-2) = 13(4)
q=52
Hence, required polynomial is:
x² - 15x + 52
Hope you Helps:)
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