Question

Find the zeros of the polynomial 4 x square - 3 x minus 1 by factorization method and verify the relationship between the zeros of the coefficients of the polynomials.

Answer 1

${\bold{\underline{\underline{Answer:}}}}$

${\bold{\therefore x=1\:and\:\frac{-1}{4}}}$

${\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\underline \bold{Given : } \\ \implies x \in({4x}^{2} - 3x - 1 = 0) \\ \\ \underline \bold{To \: Find : } \\ \implies x = ?$

• According to given question :

$\bold{Middle \: term \: spliting : } \\ \implies {4x}^{2} - 3x - 1 = 0 \\ \\ \implies {4x}^{2} - 4x + x - 1 = 0 \\ \\ \implies 4x(x - 1) + 1(x - 1) = 0 \\ \\ \implies (x - 1)(4x + 1) = 0 \\ \\ \bold{ \implies x = 1 \: and \: \frac{ - 1}{4} } \\ \\ \bold{Relationship : } \\ \implies Sum \: of \: zeroes = \frac{ - b}{a} \\ \\ \implies \frac{4 -1}{4} = \frac{ - ( - 3)}{4} \\ \\ \implies \frac{3}{4} = \frac{3}{4} \\\: \: \: \: \: \: \: \: \: \: \bold{Verified }\\ \\ \implies Product \: of \: zeroes= \frac{c}{a} \\ \\ \implies 1 \times \frac{ - 1}{4} = \frac{ - 1}{4} \\ \\ \implies \frac{ - 1}{4} = \frac{ - 1}{4} \\ \: \: \: \: \: \: \: \: \: \: \bold{Verified }$

Answer 2

$\huge{\mathfrak{\red{\underline{Answer :-}}}}$

★ Given :-

Polynomial: 4x² - 3x - 1

By splitting the middle term

4x² - 4x + x - 1

4x(x - 1) + 1(x - 1)

x - 1 = 0 ; 4x + 1 = 0

x = 1 and x = -1/4

Hence,

Zeroes are 1 and -1/4

$\rule{200}{2}$

Verification :-

a = 4

b = -3

c = -1

We know that,

Sum of zeroes = -b/a = -(-3/4) = 3/4

Product of zeroes = c/a = -1/4

Verified

$\rule{200}{2}$