Question

3x3 x square minus x minus 4 find the zeros of the polynomial environment the relationship between the zeros and the coefficients ​

Answer 1

$\large\bf\underbrace{ \: Correct \: Question:-}$

find the zeroes of the polynomial 3x² - x - 4 and also find the relationship between the zeroes and coefficients.

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

• p(x) = 3x² - x -4

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

• zeroes of given polynomial
• Relationship between the zeroes and coefficients.

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

• ≫ p(x) = 3x² - x - 4

➝ 3x² - x - 4

➝ 3x² + 3x - 4x - 4

➝ 3x( x + 1) - 4(x + 1)

➝ (3x -4)(x+1)

➝ x = 4/3 or x = -1

$\underline{ \bf \dag \: Verification : - }$

Relationship between the zeroes and coefficients:-

• p(x) = 3x² - x - 4
• a = 3
• b = -1
• c = -4

≫ Sum of zeroes = -b/a

»» 4/3 +(-1) = -(-1)/3

»» 4/3 -1 = 1/3

»» (4-3)/3 = 1/3

»» 1/3 = 1/3

≫ Product of zeroes = c/a

»» 4/3 × (-1) = -4/3

»» -4/3 = -4/3

LHS = RHS

hence , relationship is verified.

Answer 2

$\large\mathrm\red{Given:-}$

• p(x) = 3x² - x - 4

$\large\mathrm\red{Solution}$

$\implies$p(x) = 3x² - x - 4

$\implies$3x² - (4 - 3)x - 4

$\implies$3x² - 4x + 3x - 4

$\implies$x(3x - 4) + (3x - 4)

$\implies$(3x - 4) (x - 1)

$\large\mathrm\red{Now}$,

$\implies$3x - 4 = 0

$\implies$3x = 4

$\implies$x = 4/3

$\large\mathrm\red{And}$,

$\implies$x + 1 = 0

$\implies$x = - 1

The two zeroes of p(x) are 4/3 and - 1.

$\large\mathrm\red{We \: know \: that,}$

$\large\mathrm\red{Sum \: of \: zeroes }$(α + β)

$\implies$4/3 + (-1)

$\implies$4/3 - 1

$\implies$4 - 3 / 3

$\implies$1/3

$\large\mathrm\red{So}$,

• - co -efficient of x / co -efficient of x² = 1/3. $\large\mathrm\red{(Sum \: of \: zeroes)}$

$\large\mathrm\red{Product \: of \: zeroes}$(α.β)

$\implies$4/3 × (-1)

$\implies$- 4/3

$\large\mathrm\red{So}$,

• constant/ co -efficient of x² = - 4/3. $\large\mathrm\red{(product \: of \: zeroes).}$

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