Question

Find the zeros and verify the relationship between the zeros and their cofficient under root 3xsquare plus 10xsquareplus7under root 3

Answer 1

$\bold{\blue{\underline{\red{G}\pink{iv}\green{en}\purple{:-}}}}$

• P(x) = √3x² + 10x + 7√3

$\bold{\blue{\underline{\red{Zeros}\pink{\:of}\green{\:the\:polynomials}\purple{:-}}}}$

• Zeros of the polynomial is the value of the variable for which when replaced in the polynomial the value for the polynomial changes to zero.

$\bold{\blue{\underline{\red{General}\pink{\:form \:of }\green{\:polynomial}\purple{:-}}}}$

• $\bold{a_{n}x^n + a_{n-1}x^{n-1} +...... + a_{2}x^2 + a_{1}x + a_{0}}$ Where ' $\bold{a_{n} , a_{n-1} ...... a_{1} , a_{0}}$ ' are Real numbers and 'n' is an integer

.$\bold{\blue{\underline{\red{Q}\pink{uest}\green{ion}\purple{:-}}}}$

• Find the zeros and verify the relationship between the zeros and their coefficient 'P(x) = √3x² + 10x + 7√3' ?

$\bold{\blue{\underline{\red{S}\pink{olut}\green{ion}\purple{:-}}}}$  

   $\bold{\blue{\underline{\red{To}\pink{\:find}\green{\:the\:zeros}\purple{:-}}}}$

• P(x) = √3x² + 10x + 7√3

• P(x) = √3x² + 7x + 3x + 7√3      

    ∵ √3 × 7√3 = 21 ⇒ m + n = 10x & m × n = 21

• P(x) = x(√3x + 7 ) + √3²x + 7

• P(x) = x(√3x + 7 ) + √3(√3x + 7)

• P(x) = (x + √3)(√3 + 7)

• x + √3 = 0

• x = -√3

    $\\\boxed{\bold{\alpha = -\sqrt{3}}}$

• √3x + 7 = 0

• x = $\bold{\frac{-7}{\sqrt{3} } }$

    $\boxed{\bold{\beta\:=\:{\bold{\frac{-7}{\sqrt{3} }}}}}$

    $\bold{\blue{\underline{\red{V}\pink{eri}\green{fication}\purple{:-}}}}$

• α + β = $\bold{\frac{-b}{a} }$  

• -√3 + $\bold{\frac{-7}{\sqrt{3} } }$  = $\bold{\frac{-10}{\sqrt{3} } }$

• $\bold{\frac{-7-3}{\sqrt{3} } }$  = $\bold{\frac{-10}{\sqrt{3} } }$

    $\boxed{\bold{\frac{-10}{\sqrt{3} } } = \bold{\frac{-10}{\sqrt{3} } } }$

• αβ = $\bold{\frac{c}{a} }$

• -√3 × $\bold{\frac{-7}{\sqrt{3} } }$  = $\bold{\frac{7\sqrt{3} }{\sqrt{3} } }$

    $\boxed{\bold{7\:=\:7}}$