Question

Find the zeros of the polynomial x²-7x+12 and find the relationship between the zeroes and coefficients?​

Answer 1

Solution

Given :-

• Polynomial equation, x² - 7x +12 = 0

Find :-

• Zeroes of this Equation,
• Relationship between zeroes & uts coefficient

Explanation

Let,

• Zeroes be p & q

Using Factories Method .

==> x² - 7x + 12 = 0

==> x² - 4x - 3x + 12 = 0

==> x(x -4) - 3(x - 4) = 0

==> (x - 3)(x - 4) = 0

==> (x - 3) = 0. Or, x - 4 = 0

==> x = 3 Or, x = 4

Since,

• Zeroes be ,p = 3
• q = 4.

________________________

Now, Calculate relationship between zeroes & coefficient.

Using Formula

★ Sum of zeroes = -(Coefficient of x)/(coefficient of x²)

==> p + q = -(-7)/1

==> p + q = 7___________(1)

keep value of p & q,

==> 3 + 4 = 7

==> 7 = 7

L.H.S. = R.H.S.

__________________

Again,

★ Product of zeroes = (constant part)/(coefficient of x²)

==> p.q = 12/1

==> p.q = 12 __________(2)

keep value of p & q

==> 3 × 4 = 12

==> 12 = 12

L.H.S. = R.H.S.

That's Proved.

________________

Answer 2

Given Polynomial : x² - 7x + 12

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Need to find : The Zeroes of the given polynomial & also we've to verify the relationship b/w its zeroes & their cofficients.

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$\:\:$❭❭ Finding out the zeroes of Given polynomial :

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$\:\:\:\:\:\:\:\:\:\::\:\Longrightarrow\:{\sf{x²\:-\:7x\:+\:12}}$

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$\:\:\:\:\:\:\:\::\:\Longrightarrow\:{\sf{x²\:-\:7x\:+\:12\:=\:0}}$

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$\:\:\:\:\:\::\:\Longrightarrow\:{\sf{x²\:-\:4x\:-\:3x\:+\:12\:=\:0}}$

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$\:\:\:\:\:\::\:\Longrightarrow\:{\sf{(x\:-\:3)\:(x\:-\:4)}}$

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$\:\:\:\:\:\:\:\::\:\Longrightarrow\:{\underline{\boxed{\frak{\blue{x\:=\:3\:or\:x\:=\:4}}}}}\:\bigstar$

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$\:\therefore\:{\underline{\sf{Hence,\:the\:zeroes\:of\:the\:given\:polynomial\:are\:{\textsf{\textbf{3\:and\:4 }}}}}}.$

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$\:\:\:\:\:\:\:\:━━━━━━━━━━━━━━━━━━━$

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V E R I F I C A T I O N :

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☆ If α and β are roots of any Quadratic equation ax² + bx + c = 0 then Sum and Product is given by :

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$\:\:\:\:\:{\bold{⋆}}\:$Sum (α + β) = (-b)/ a

$\:\:\:\:\:{\bold{⋆}}\:$Product (αβ) = c/a

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$\:\:\:\:\:\:\:\:\:\:\:\:\:{\sf{✠\:Sum\:of\:Zeroes\::}}$

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$\:\:\:\:\:\:\:\:\:{↠\:\alpha\:+\:\beta\:=\:\sf{\dfrac{-b}{a}}}$

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$\:\:\sf{↠\:(3)\:+\:(4)\:=\:{\dfrac{-\:(-7)}{1}}}$

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$\:\:\:\:\:\:\:\:\:\sf{↠\:7\:=\:-\:(-7)}$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:↠\:{\underline{\boxed{\frak{7\:=\:7}}}}$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:{\sf{✠\:Product\:of\:Zeroes\::}}$

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$\:\:\:\:\:\:\:\:\:{↠\:\alpha\:\beta\:=\:\sf{\dfrac{c}{a}}}$

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$\:\:\sf{↠\:(3)\:\times\:(4)\:=\:{\dfrac{12}{1}}}$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:↠\:{\underline{\boxed{\frak{12\:=\:12}}}}$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:\therefore\:{\underline{\purple{\textsf{\textbf{Hence,\:Verified!}}}}}$

$\:\:\:\:\:\:\:\:━━━━━━━━━━━━━━━━━━━$

${\sf{\bf{\red{@ℐᴛᴢӇᴀᴘᴘʏҨᴜᴇᴇɴ࿐}}}}$