if a and b are the zeros of the quadratic polynomial f(x) = 2x square - 5x + 7 ,find the polynomial whose zeros are 2a + 3 b and 3a + 2b
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f(x) = 2x²-5x+7
a and b are the zeroes.
→ Sum of zeroes = a+b = 5/2
→ Product of zeroes = ab = 7/2
To find quadratic polynomial,
Given that,
2a+3b and 3a+2b are the zeroes of the polynomial.
Sum of zeroes = 2a+3b+3a+2b
= 5a+5b
=5(a+b)
=5(5/2)
=25/2
Product of zeroes = ab = (2a+3b)(3a+2b)
= 6a²+4ab+9ab+6b²
= 6(a²+b²)+13ab
a²+b² = (a+b)²-2ab
= 6{(5/2)²-2(7/2)} + 13(7/2)
= 6(25/4-7) + 91/2
= 6(-3/4) + 91/2
= -9/2 + 91/2
= (-9+91)/2
= 82/2
= 41
Therefore, the quadratic polynomial is expressed in the form of
k{x² - (sum of zeroes)x + (product of zeroes)}
→ k{x²-25x/2+41}
Put k = 2,
→ 2x²-25x+82
Hope it helps
a and b are the zeroes.
→ Sum of zeroes = a+b = 5/2
→ Product of zeroes = ab = 7/2
To find quadratic polynomial,
Given that,
2a+3b and 3a+2b are the zeroes of the polynomial.
Sum of zeroes = 2a+3b+3a+2b
= 5a+5b
=5(a+b)
=5(5/2)
=25/2
Product of zeroes = ab = (2a+3b)(3a+2b)
= 6a²+4ab+9ab+6b²
= 6(a²+b²)+13ab
a²+b² = (a+b)²-2ab
= 6{(5/2)²-2(7/2)} + 13(7/2)
= 6(25/4-7) + 91/2
= 6(-3/4) + 91/2
= -9/2 + 91/2
= (-9+91)/2
= 82/2
= 41
Therefore, the quadratic polynomial is expressed in the form of
k{x² - (sum of zeroes)x + (product of zeroes)}
→ k{x²-25x/2+41}
Put k = 2,
→ 2x²-25x+82
Hope it helps
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