Question

if alpha and beta are the zeros of the polynomial x2-2x-15, then form a quadratic polynomial whose zeros are 2alphaand 2beta.

Answer 1

since alpha and beta are the zeroes of given polynomial x^2-2x-15
=x^2-2x-15
=x^2-5x+3x-15
=x(x-5)+3(x-5)
=(x-5)(x+3)
=. x=5&x=-3
therefore alpha=5β=-3
according to question
zeroes of other polynomial are alpha' =2alpha & beta'=2beta
therefore alpha'=10 & beta'= -6
polynomial=x^2-(10-6)x+10×-6
=x^2-4x-60

Answer 2

Answer:

The new polynomial is x² - 4x - 60

Step-by-step explanation:

Given polynomial = x² - 2x - 15

The given polynomial's zero are = α and β

The new polynomial when zeros are 2α and 2β = ??

In polynomial :

• a = 1
• b = -2
• c = -15

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Sum of zeros:

$\sf{\longrightarrow}\: \alpha + \beta =\dfrac{-b}{a} \\\\\sf{\longrightarrow}\: \alpha + \beta =\dfrac{-(-2)}{1}\\\\\sf{\longrightarrow}\: \alpha + \beta =\dfrac{2}{1}\\\\\sf{\longrightarrow}\: \alpha + \beta =2 \:---(I)$

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Product of zeros:

$\sf{\longrightarrow}\: \alpha \times \beta = \dfrac{c}{a} \\\\\sf{\longrightarrow}\: \alpha \times \beta = \dfrac{-15}{1} \\\\\sf{\longrightarrow}\: \alpha \times \beta = -15 \: ---(II)$

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For the new polynomial,

$\sf{\longrightarrow}\:2\alpha +2\beta \\\\\sf{\longrightarrow}\:2(\alpha +\beta)\\\\\sf{\longrightarrow}\:2(2)\: ---(From\:I)\\\\\sf{\longrightarrow}\:4$

The sum of zeros for new polynomial is 4.

$\sf{\longrightarrow}\: 2\alpha \times 2\beta \\\\\sf{\longrightarrow}\: 4 \times \alpha\beta \\\\\sf{\longrightarrow}\: 4 \times (-15)\:---(From\:II)\\\\\sf{\longrightarrow}\: -60$

The product of zeros for new polynomial is -60.

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The new polynomial will be =

$\sf{\longrightarrow}\: x^{2} - (\alpha +\beta )x + (\alpha \beta )\\\\\sf{\longrightarrow}\: x^{2} - (4 )x + (-60)\\\\\sf{\longrightarrow}\: x^{2} - 4x-60$

Therefore, the new polynomial is x² - 4x - 60