Question

find the zeros of the polynomial and verify the relationship between the zeros and their cofficients of tsquare-121​

Answer 1

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

We have polynomial t² - 121.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The zeroes and verify the relationship between the zeroes and coefficient.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

p(x) = t² - 121

Zero of the polynomial is p(x) = 0

So;

$\longrightarrow\sf{t^{2} -121=0}\\\\\longrightarrow\sf{t^{2} =121}\\\\\longrightarrow\sf{t=\pm\sqrt{121} }\\\\\longrightarrow\sf{t=\pm11}$

∴ The α = 11 and β = -11 are the zeroes of the polynomial.

As the given quadratic polynomial as we compared with ax² + bx + c

• a= 1

• b = 0

• c = -121

Now;

$\underline{\green{\mathcal{SUM\:OF\:THE\:ZEROES\::}}}$

$\mapsto\sf{\alpha +\beta =\dfrac{-b}{a} =\dfrac{Coefficient\:of\:x}{Coefficient\:of\:(x)^{2} } }\\\\\\\mapsto\sf{11 +(-11) =\dfrac{-0}{1} }\\\\\\\mapsto\sf{11-11=0}\\\\\\\mapsto\sf{\red{0=0}}$

$\underline{\green{\mathcal{PRODUCT\:OF\:THE\:ZEROES\::}}}$

$\mapsto\sf{\alpha \times \beta =\dfrac{c}{a} =\dfrac{Constant\:term}{Coefficient\:of\:(x)^{2} } }\\\\\\\mapsto\sf{11 \times (-11) =\dfrac{-121}{1} }\\\\\\\mapsto\sf{\red{-121=-121}}$

Thus;

Relationship between zeroes and coefficient is verified .

Answer 2

Answer:

⋆ Given Polynomial : t² – 121

Here : a = 1,⠀b = 0,⠀c = – 121

$:\implies\tt f(x) = 0\\\\\\:\implies\tt t^2-121= 0\\\\\\:\implies\tt (t)^2-(11)^2= 0\\\\{\scriptsize\qquad\bf{\dag}\:\:\sf{(a)^2-(b)^2=(a+b)(a-b)}}\\\\:\implies\tt(t+11)(t-11)=0\\\\\\:\implies\tt\underline{\boxed{\tt t =-\:11\quad or\quad t=11}}$

$\rule{150}{1}$

$\underline{\bigstar\:\textsf{Relation b/w zeroes and coefficient :}}$

$\qquad\underline{\bf{\dag}\:\:\textsf{Sum of Zeroes :}}\\\dashrightarrow\rm\:\: \alpha+\beta = \dfrac{-\:b}{a}\\\\\\\dashrightarrow\rm\:\: -\:11+11= \dfrac{-0}{1}\\\\\\\dashrightarrow\:\:\underline{\boxed{\red{\rm 0=0}}}\\\\\\{\qquad\underline{\bf{\dag}\:\:\textsf{Product of Zeroes :}}}\\\\\dashrightarrow\rm\:\: \alpha \times \beta = \dfrac{c}{a}\\\\\\\dashrightarrow\rm\:\: -\:11\times11 = \dfrac{-121}{1}\\\\\\\dashrightarrow\:\:\underline{\boxed{ \red{\rm -\:121=-\:121}}} \\\\{\qquad\underline{\mathscr{VERIFIED}}}$