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Polynomials

Long Answer Type

421. Find a quadratic polynomial whose zeroes are 1 and (-3). Verify the relation between the coefficients and zeroes of the polynomial.

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422. Find the zeroes of the quadratic polynomial 4x² – 4x – 3 and verify the relation between the zeroes and its coefficients.

4x² – 4x – 3
= 4x² – 6x + 2x – 3
= 2x (2x – 3) + 1 (2x – 3)
= (2x – 3) (2x + 1)
So, 4x² – 4x – 3 = 0 ⇒ (2x – 3) (2x + 1) = 0
⇒ 2x – 3 = 0 or 2x + 1 = 0
⇒ x = 3/2 or x = – 1/2
∴ Zeroes of $4 straight x squared minus 4 straight x minus 3$ are $3 over 2 space and space minus 1 half$
Now,
Sum of zeroes = $3 over 2 plus open parentheses negative 1 half close parentheses space equals space 1 equals fraction numerator negative left parenthesis negative 4 right parenthesis over denominator 4 end fraction equals negative straight b over straight a$

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and product of zeroes = $3 over 2 cross times open parentheses negative 1 half close parentheses equals fraction numerator negative 3 over denominator 4 end fraction equals straight c over straight a$

$equals space fraction numerator Constant space term over denominator Coefficient space of space straight x squared end fraction$