Question

If alpha and beta are the zeroes of polynomial p(x) = x² - 7x + 10, find the quadratic polynomial with zeroes (-alpha) and (-beta).​

Answer 1

$\footnotesize{\underline{\textsf{\textbf{\red{G\:I\:V\:E\:N}}}}}$

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• α and β are the zeroes of polynomial p(x) = x² - 7x + 10.

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$\footnotesize{\underline{\textsf{\textbf{\green{T\:O\:\:\:F\:I\:N\:D}}}}}$

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• Quadratic polynomial with zeroes -α and -β.

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$\footnotesize{\underline{\textsf{\textbf{\blue{S\:O\:L\:U\:T\:I\:O\:N}}}}}$

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Finding zeroes of polynomial x² - 7x + 10 ::

$\\ \dashrightarrow \: \sf x^2 - 7x + 10 = 0$

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Splitting the middle term ::

$\\ \dashrightarrow \: \sf x^2 - 5x - 2x + 10 = 0$

$\\ \dashrightarrow \: \sf x(x - 5) - 2(x - 5) = 0$

$\\ \dashrightarrow \: \sf (x - 5)\:(x - 2) = 0$

$\\ \dashrightarrow \: \sf x - 5 = 0,\: x - 2 = 0$

$\\ \dashrightarrow \: \sf x = 0 + 5,\: x = 0 + 2$

$\\ \dashrightarrow \: \bf \pink{x = 5,\: x = 2}$

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• Hence, zeroes of polynomial x² - 7x + 10 (α and β) are 5 and 2.

• So, zeroes of required polynomial (-α and -β) are -5 and -2.

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Finding sum of zeroes (S) and product of zeroes (P) of required polynomial ::

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➝ Sum of zeroes (S) = -5 + (-2)

➝ Sum of zeroes (S) = - 5 - 2

➝ Sum of zeroes (S) = -7

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➝ Product of zeroes (P) = (-5) × (-2)

➝ Product of zeroes (P) = 10

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Finding the polynomial ::

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$\footnotesize{\underline{\boxed{\sf{Eq^{n}\:for\:finding\:any\:quadratic\:polynomial\:is\:\bf{x^2 - Sx + P}}}}}$

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Putting values of 'S' and 'P' in eqⁿ ::

$\\ \dashrightarrow \: \sf x^2 - (-7)x + 10$

$\\ \dashrightarrow \: \bf \purple{x^2 + 7x + 10}$

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$\therefore\:{\underline{\bf{x^2 + 7x + 10\:\sf{is\:the\:required\:polynomial!}}}}$

$\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare$

Answer 2

x² + 7x + 10

Step-by-step explanation:

QUESTION :-

If alpha and beta are zeroes of polynomial p(x) = x² - 7x + 10, find the quadratic polynomial with zeroes (- alpha) and (- beta).

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SOLUTION :-

p(x) = x² - 7x + 10

In the standard form of quadratic polynomial (ax² + bx + c), here -

• a = 1
• b = -7
• c = 10

We know that,

sum of zeroes (α+β) = - b/a

= -(-7)/1

= 7

Product of zeroes (αβ) = c/a

= 10/1

= 10

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Now,

we have to find the quadratic polynomial whose zeroes are (-α) and (-β)

So,

sum of their zeroes (S) = (-α) + (-β)

= -α-β

= -(α + β)

= -(7)

= -7

Product of their zeroes (P) = (-α)(-β)

= αβ

= 10

To find the quadratic polynomial, we use formula

x² -(S)x + P

=> x² -(-7)x + 10

=> x² + 7x + 10

So,

The quadratic polynomial is x² + 7x + 10.

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Hope it helps.