Question

Find zeros of polynomial 2x^2-3x+1 and verify the relationship between coefficients and zeros of polynomial​

Answer 1

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{Zeroes \ of \ the \ polynomial \ are \ 1 \ and \ \frac{1}{2}}$

$\sf\orange{Given:}$

$\sf{The \ given \ quadratic \ polynomial \ is}$

$\sf{\implies{2x^{2}-3x+1}}$

$\sf\pink{To \ find:}$

$\sf{Zeroes \ of \ the \ polynomial. }$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{The \ given \ quadratic \ polynomial \ is}$

$\sf{\implies{2x^{2}-3x+1}}$

$\sf{\implies{2x^{2}-2x-x+1}}$

$\sf{\implies{2x(x-1)-1(x-1)}}$

$\sf{\implies{(x-1)(2x-1)}}$

$\sf{\implies{\therefore{x=1 \ or \ \frac{1}{2}}}}$

$\sf{Zeroes \ of \ the \ polynomial \ are \ 1 \ and \ \frac{1}{2}}$

_______________________________________

$\blue{\underline{\underline{Verification:}}}$

$\sf{The \ given \ quadratic \ polynomial \ is}$

$\sf{\implies{2x^{2}-3x+1}}$

$\sf{Here, \ a=2, b=-3 \ and \ c=1}$

$\sf{Let \ \alpha \ be \ 1 \ and \ \beta \ be \ \frac{1}{2}}$

_________________________

$\sf{\alpha+\beta=1+\frac{1}{2}}$

$\sf{\therefore{\alpha+\beta=\frac{3}{2}...(1)}}$

$\sf{\frac{-b}{a}=\frac{-(-3)}{2}}$

$\sf{\therefore{\frac{-b}{a}=\frac{3}{2}...(2)}}$

$\sf{...from \ (1) \ and \ (2)}$

$\sf{Sum \ of \ zeroes=\frac{-b}{a}}$

_________________________

$\sf{\alpha\beta=1\times\frac{1}{2}}$

$\sf{\therefore{\alpha\beta=\frac{1}{2}...(3)}}$

$\sf{\frac{c}{a}=\frac{1}{2}...(4)}$

$\sf{...from \ (3) \ and \ (4)}$

$\sf{Product \ of \ zeroes=\frac{c}{a}}$

$\sf{Hence, \ verified. }$

Answer 2

$\large\bf\underline \orange{Given:-}$

• p(x) = 2x² - 3x +1

$\large\bf\underline \orange{To \: find:-}$

• zeroes of polynomial

• relationship between the zeroes and coefficients.

$\huge\bf\underline \green{Solution:-}$

• p(x) = 2x² - 3x +1

$: \implies \rm\: {2x}^{2} - 3x + 1 \\ \\ : \implies \rm\: 2 {x}^{2} - 2x - x + 1 \\ \\ : \implies \rm\:2x(x - 1) - 1(x - 1) \\ \\ : \implies \rm\:(2x - 1)(x - 1) \\ \\ : \implies \bf\:x = \frac{1}{2} \: \: or \: \: x = 1$

• Let α be 1/2
• and β be 1

p(x) = 2x² - 3x +1

• a = 2
• b = -3
• c = 1

$\bf \underline{Verification:-}$

$: \bigstar \bf\: \alpha + \beta = \frac{ - b}{a} \\ \\ : \implies \rm\: \frac{1}{2} + 1 = \frac{ - ( -3 )}{2} \\ \\ : \implies \rm\: \frac{1 + 2}{2} = \frac{3}{2} \\ \\ : \implies \rm\: \frac{3}{2} = \frac{3}{2}$

$: \bigstar \bf\: \alpha \beta = \frac{c}{a} \\ \\ : \implies \rm\: \frac{1}{2} \times 1 = \frac{1}{2} \\ \\ : \implies \rm\: \frac{1}{2} = \frac{1}{2}$

So, LHS = RHS

hence, relationship between the zeroes and coefficients is Verified.