Question

Find the zero polynomial and verify the relationship between zero and Coefficient 2 x square minus x minus 1

Answer 1

Given :-

A polynomial

$\sf\:\: 2x{}^{2}-x-1$

To Find :-

The zero's of polynomial

A.T.Q :-

$\Longrightarrow \sf 2x^2 - x - 1 = 0\\ \\\Longrightarrow \sf 2x^2 - (2 - 1)x - 1 = 0\\ \\\Longrightarrow \sf 2x^2 - 2x + x - 1 = 0\\ \\\Longrightarrow \sf 2x(x - 1) + 1(x - 1) = 0\\ \\\Longrightarrow \sf (2x + 1)(x - 1) = 0$

• Therefore the zeroes of polynomial is 1 &-1/2

Let 1 be α and let -1/2 be β.

$\sf We \ know \ that, \\ \\\sf a \longmapsto Coefficient \ of \ x^2.\\ \\\sf b \longmapsto Coefficient \ of \ x.\\ \\\sf c \longmapsto Constant \ term.\\ \\ \\\sf We \ have,\\ \\\sf a \longmapsto 2.\\ \\\sf b \longmapsto-1.\\ \\\sf c \longmapsto -1$

$\sf Sum \ of \ zeroes \ (\alpha + \beta) = \dfrac{- b}{ \ a} \\ \\ \\\implies \sf \alpha + \beta = \dfrac{Coefficient \ of \ x}{ \ Coefficient \ of \ x^2}\\ \\ \\\implies \sf 1 + \left(-\dfrac{1}{2}\right) = \dfrac{-(-1)}{2}\\ \\ \\\implies \sf \left(\dfrac{2 - 1}{2}\right) = \dfrac{1}{2}\\ \\ \\\implies \sf \dfrac{1}{2} = \dfrac{1}{2}\\ \\ \\\implies \sf LHS = RHS\\ \\ \\\sf Hence \ Verified!$

$\sf Product \ of \ zeroes \ (\alpha \beta) = \dfrac{c}{a} \\ \\ \\\implies \sf \alpha \beta = \dfrac{Constant \ term}{Coefficient \ of \ x^2}\\ \\ \\\implies \sf 1 \times \left(-\dfrac{1}{2}\right) = \dfrac{- 1}{ \ 2}\\ \\ \\\implies \sf \dfrac{-1}{ \ 2} = \dfrac{-1}{2}\\ \\ \\ \implies \sf LHS = RHS\\ \\ \\\sf Hence \ Verified!$

Answer 2

Answer:-

Zeros of polynomial = -1/2 & 1

$\rule{200}{1}$

Given polynomial:-

☛ 2x² - x - 1 = 0

We can find zeros by factorization method:-

→ 2x² - x - 1 = 0

→ 2x² - 2x + x - 1 = 0

→ 2x(x - 1) + 1(x - 1) = 0

→ (2x + 1)(x - 1) = 0

→ 2x = -1 or x = 1

→ x = -1/2 or x = 1

Hence,

$\therefore\underline{\textsf{Zeros of polynomial = {\textbf{ -1/2 \& 1 }}}}$

$\rule{200}{1}$

Verification:-

Here, zeros are -1/2 & 1

Comparing the given polynomial with ax² + bx + c we get:-

• Coefficient of x²(a) = 2
• Coefficient of x(b) = -1
• Constant term(c) = -1

Relationship 1:-

☛ Sum of zeros = -b/a

→ (-1/2 + 1) = -(-1)/2

→ (-1 + 2)/2 = 1/2

→ 1/2 = 1/2 [Hence verified!]

Relationship 2:-

☛ Product of zeros = c/a

→ (-1/2 × 1) = -1/2

→ -1/2 = -1/2 [Hence verified!]

Therefore,

Relation b/w zeros & coefficients of polynomial are verified.