Question

Find the zeros of the quadratic polynomial X²-5 and verify the relationship between the zeros and coefficients

Answer 1

Answer:

• Zeroes of polynomial are √5 and -√5.

Step by step explanation:

Given:

• Polynomial - x² - 5

To Find:

• Zero of the polynomial.
• Verify relationship between zeroes and coefficient.

Now, first we will find zero of the polynomial.

Let zero of the polynomial be α and β.

=> x² - 5 = 0

=> x² = 5

=> x = ±√5

Hence, zeroes of polynomial are +√5 and -√5.

Here,

• α = √5
• β = -√5

Verification:

=> Sum of zeroes = -b/a

=> √5 - √5 = 0/1

=> 0 = 0

=> Product of zeroes = c/a

=> (√5)(-√5) = -5/1

=> -5 = -5/1

Hence Proved!!

Answer 2

• The Zeroes of any Polynomials are the real values which makes the polynomial zero.
• The Zeroes and degree( highest power ) of a polynomial are co-related. Degree Of a Polynomial = Number Of Zeroes.

Since, we have a quadratic polynomial which has degree 2, therefore it will also have 2 zeroes.

NOW,

$\mathsf { P(x) \: = \: x^2 \: - \: 5 }$

Now,

$\mathsf{ \implies {x}^{2} \: = \: 5}$

$\mathsf {\therefore \: \: x \: = \: \pm \sqrt{5}}$

Now,

Coefficient of $\mathsf{x^{2} }$ (a) = 1

Coefficient of x (b) = 0

Constant term (c) = -5

And,

$\mathsf{ \implies \: Sum \: of \: zeroes \: = \: \dfrac{-b}{a}}$

   

$\mathsf{ \implies \: \sqrt{5} \: - \: \sqrt{5} \: = \: \dfrac{-0}{1}}$

$\mathsf{\therefore \: \: 0 \: = \: 0 }$

And,

$\mathsf{ \implies \: Product \: of \: zeroes \: = \: \dfrac{c}{a}}$

$\mathsf{ \implies \: \sqrt{5} \: \cdot \: (-\sqrt{5}) \: = \: \dfrac{-5}{1}}$

$\mathsf{\therefore \: \: -5 \: = \: -5 }$

Verified !!