Question

PLEASE ANSWER If α and β are zeros the polynomial p(x) = 3x² - 5x + 7, then find the value of α² + β².

Answer 1

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

• p(x) = 3x² - 5x + 7

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

• Value of α² + β²

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

• p(x) = 3x² - 5x +7

• a = 3
• b = -5
• c = 7

Let α and β are the zeroes of the given polynomial.

≫ Sum of zeroes = -b/a

⠀⠀⠀⠀⠀» α + β = -(-5)/3

⠀⠀⠀⠀⠀» α + β = 5/3

≫ Product of zeroes = c/a

⠀⠀⠀⠀⠀» αβ = 7/3

we know that,

⠀⠀⠀⠀⠀»» (a+b)² = a² + b² + 2ab

⠀⠀⠀⠀⠀»» (a+b)² - 2ab = a² + b²

So, α² + β² = (α + β)² - 2αβ .....(i)

Putting values of α + β = 5/3 and αβ = 7/3 in equation (i) .

⠀⠀⠀⠀⠀➝ α² + β² = (5/3)² - 2× 7/3

⠀⠀⠀⠀⠀➝ α² + β² = 25/9 -14/3

⠀⠀⠀⠀⠀➝ α² + β² = (25-42)/9

⠀⠀⠀⠀⠀➝ α² + β² = -17/9

So, value of α² + β² = -17/9.

Answer 2

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

If α and β are zeros the polynomial p(x) = 3x² - 5x + 7, then find the value of α² + β².

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

• $\tt\alpha\:and\:\beta\:are\:root\:of\:the\: polynomial$
• 3x² - 5x + 7

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

• value of α² + β².

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

$\tt→ 3x^2-5x+7\\\tt→ x=3\\\tt→ y= -5\\\tt→ z= 7$

Let,

x and y are the root of the given polynomial.

So,

sum of zeroes → $\tt \frac{-y}{x}$

$\tt→ x + y = -\frac{-5}{3}\\\tt→ x+y=\frac{5}{3}$

Product of zeroes = xy

$\tt→ xy= \frac{z}{x}\\\tt→ xy=\frac{7}{3}$

$\bf\huge Now,\\\tt→ (x+y)^2=x^2+2xy+y^3\\\tt→ (x+y)^2-2xy=x^2+y^2\\\bf\huge Then,\\\tt→ x^2+y^2 = (x+y)^2-2xy--equ(1)$

Now,

substitute the value of x+y and xy in equ(1)

$\tt\maltese x^2+y^2= (\frac{5}{3})^2-2(\frac{7}{3})\\\tt\maltese x^2+y^2= \frac{25}{9}-\frac{14}{3}\\\tt\maltese x^2+y^2=\frac{25-42}{9}\\\tt\maltese x^2+y^2= \frac{-17}{9}$

$\bf\huge{\underline{\red{\underline{Hence:-}}}}$

$\tt→{\fbox{\red{\fbox{x^2+y^2=\frac{-17}{9}}}}}$