Question

If alpha and beta are the zeroes of the quadratic polynomial : x²-7x+10, evaluate alpha³+beta³​

Answer 1

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

• we need to find the value of α³ + β³

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

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

• Quadratic polynomial :- x² - 7x + 10
• a = 1
• b = -7
• c = 10

we know that,

⚘ Sum of zeroes = -b/a

↛ α + β = -(-7)/1

↛ α + β = 7 .....1)

⚘ Product of zeroes = c/a

↛ αβ = 10/1

↛ αβ = 10 .......2)

Now,

we know that,

⇝ (a + b)² = a² + b² + 2ab

⇝ a² + b² = (a + b)² - 2ab ......3)

Now,

• Value of α³ + β³ is :-

⇝ α³ + β³ = (α + β)(α² + β² - αβ)

• From 1) , 2) and 3)

⇝ α³ + β³ = 7[(α + β)² - 2αβ - αβ]

⇝ α³ + β³ = 7 [7² - 2 × 10 - 10]

⇝ α³ + β³ = 7(49 - 20 - 10)

⇝ α³ + β³ = 7( 49 - 30)

⇝ α³ + β³ = 7 × 19

⇝ α³ + β³ = 133

Hence,

• Value of α³ + β³ is 133

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Answer 2

$\mathbb{\huge{\fbox{\fbox{\red{QUESTION\:?}}}}}$

✴ If alpha and beta are the zeroes of the quadratic polynomial : x²-7x+10. Evaluate alpha³+beta³ .

$\mathcal{\huge{\fbox{\green{AnSwEr:-}}}}$

➡ α³ + β³ is 133

$\mathcal{\huge{\fbox{\purple{Solution:-}}}}$

Given, quadratic polynomial :- x² - 7x + 10

Taking,

• a = 1
• b = -7
• c = 10

• Sum of zeroes = -b/a

↗ α + β = -(-7)/1

↗ α + β = 7 .....(1)

• Product of zeroes = c/a

↗ αβ = 10/1

↗ αβ = 10 .......(2)

Now,

We know that,

▶ (a + b)² = a² + b² + 2ab

↗ a² + b² = (a + b)² - 2ab ......(3)

So, α³ + β³

▶ α³ + β³ = (α + β)(α² + β² - αβ)

Taking from (1, 2 & 3)

▶ α³ + β³

▶ 7{(α + β)² - 2αβ - αβ}

▶ 7 {7² - 2 × 10 - 10}

▶ 7(49 - 20 - 10)

▶ 7( 49 - 30)

▶ 7 × 19

▶ 133

Hence,The Value of α³ + β³ is 133.

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