Question

$if \: \alpha \: and \: \beta \: are \: the \: zeroes \: of \: the \\ quadratic \: polynomial \\ f(x) = {x}^{2} - 4x + 3 \\ find \: the \: value \: of \: { \alpha }^{4} { \beta }^{2} + { \alpha }^{2} { \beta }^{4}$
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Answer 1

f(x) = x² - 4x + 3

It's zeroes are α and β.

From relationship of zeroes and coefficients of a quadratic polynomial, we have,

• α + β = 4
• αβ = 3

Now,

$\large \implies \tt α + β = 4 \\ \tiny{ \rm \: squaring \: both \: sides} \\ \implies \tt {α}^{2} + {β}^{2} + 2αβ = 4 \\ \\ \implies \tt {α}^{2} + {β}^{2} = 4 - 2αβ \\ \tiny{ \rm putting \: the \: known \: values}\\ \implies \tt {α}^{2} + {β}^{2} = 4 - 3 \\ \\ \implies \tt {α}^{2} + {β}^{2} = 1$

Thus,

→ α⁴β² + α²β⁴

→ α²β²(α² + β²)

→ (αβ)² × (α² + β²)

→ (3)² × (1)

→ 9 × 1 = 9