Question

1. If a and B are the zeroes of the polynomial 3x² – 2x – 7, then find the value of
I. (1/alpha) + (1/beta) +2 aß.
ii. alpha2 + beta2​

Answer 1

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

• we need to find the Value of
• (i) 1/α+1/β + 2
• (ii) α² + β²

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

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

α and β are zeroes of polynomial 3x² - 2x - 7

• Polynomial = 3x² - 2x - 7
• a = 3
• b = -2
• c = -7

We know that ,

▶ Sum of zeroes = -b/a

→ α + β = -(-2)/3

→ α + β = 2/3 ....1)

▶ Product of zeroes = c/a

→ αβ = -7/3 .....2)

Now, Finding value of :-

(i) 1/α+1/β + 2

→ 1/α+1/β + 2 = (α + β)/αβ + 2

From 1) and 2)

→ 1/α+1/β + 2 = 2/3/-7/3 + 2

→ 1/α+1/β + 2 = 2/3 × -3/7 + 2

→ 1/α+1/β + 2 = -6/21 + 2

→ 1/α+1/β + 2 = (-6 + 42)21

→ 1/α+1/β + 2 = 36/21

→ 1/α+1/β + 2 = 12/7

(ii) α² + β²

we know that,

→ (α + β)² = α² + β² + 2αβ

→ α² + β² = (α + β)² - 2αβ

From 1) and 2)

→ α² + β² = (2/3)² - 2 × -7/3

→ α² + β² = 4/9 + 14/3

→ α² + β² = (4 + 42)/9

→ α² + β² = 46/9

Hence,

● Value of 1/α+1/β + 2 = 12/7

● Value of α² + β² = 46/9

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Answer 2

Step-by-step explanation:

α and β are zeroes of polynomial 3x² - 2x - 7

Polynomial = 3x² - 2x - 7

a = 3

b = -2

c = -7

We know that ,

▶ Sum of zeroes = -b/a

→ α + β = -(-2)/3

→ α + β = 2/3 ....1)

▶ Product of zeroes = c/a

→ αβ = -7/3 .....2)

Now, Finding value of :-

(i) 1/α+1/β + 2

→ 1/α+1/β + 2 = (α + β)/αβ + 2

From 1) and 2)

→ 1/α+1/β + 2 = 2/3/-7/3 + 2

→ 1/α+1/β + 2 = 2/3 × -3/7 + 2

→ 1/α+1/β + 2 = -6/21 + 2

→ 1/α+1/β + 2 = (-6 + 42)21

→ 1/α+1/β + 2 = 36/21

→ 1/α+1/β + 2 = 12/7

(ii) α² + β²

we know that,

→ (α + β)² = α² + β² + 2αβ

→ α² + β² = (α + β)² - 2αβ

From 1) and 2)

→ α² + β² = (2/3)² - 2 × -7/3

→ α² + β² = 4/9 + 14/3

→ α² + β² = (4 + 42)/9

→ α² + β² = 46/9

Hence,

● Value of 1/α+1/β + 2 = 12/7

● Value of α² + β² = 46/9

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