Question

If a and b are zeroes of the polynomial x2- x - 6, then find a quadratic polynomial whose zeroes are (3a + 2b) and (2a + 3b)?

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Answer 1

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Answer 2

Answer:

The quadratic polynomial with zeroes (3a+2b) and (2a+3b) is $x^{2} -5x$ .

Step-by-step explanation:

Given:- a and b are zeroes of $x^{2} -x-6$.

To Find:- Quadratic polynomial whose zeroes are (3a+2b) and (2a+3b)

Solution:-

⇒ $x^{2} -x-6$ = 0

⇒ $x^{2} -(3-2)x-6$ = 0

⇒ $x^{2} -3x+2x-6$ = 0

⇒ $x^{}(x-3)+2(x-3)$ = 0

⇒ $(x^{}-3) (x+2)$ = 0

∴ x = 3  or x = -2

∴ a = 3 and b = -2.

Now, as we know the expression of a quadratic polynomial is

$x^{2} -(sum of roots)x+(product of roots)$

∴ $x^{2} -(3a+2b+2a+3b)x + ((3a+2b)(2a+3b))$

= $x^{2} -(5a+5b)x+(6a^{2} +6b^{2} +13ab)$

Now, putting values of a and b, we get:

= $x^{2} - (5(3)+5(-2))x+(6(3^{2})+6(-2)^{2} +13(3)(-2))$

= $x^{2} -(15-10)x+(54+24-78)$

= $x^{2} -5x$

Hence, The quadratic polynomial with zeroes (3a+2b) and (2a+3b) is $x^{2} -5x$ .

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