Question

y+z=4 yz=3 find y³+z³=?

Answer 1

Answer:

★Question★

• If y + z = 4 and yz = 3, find y³ + z³

★Answer★

Given:-

• y + z = 4 and yz = 3

To find :-

• y³ + z³

Identity used :-

$\bf\implies \: {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)$

Solution :-

y + z = 4........(1) and yz = 3

Cubing (1) both sides

$\bf\implies \: {(y + z)}^{3} = {4}^{3}$

$\bf\implies \: {y}^{3} + {z}^{3} + 3yz(y + z) = 64$

$\bf\implies \: y³ + z³ + 3 \times 3 \times 4 = 64$

$\bf\implies \:y³ + z³ + 36 = 64$

$\bf\implies \:y³ + z³ = 64 - 36$

$\bf\implies \:y³ + z³ = 28$

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Additional Information

Other useful identities :-

$\bf\ \: {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2}$

$\bf \: {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2}$

$\bf \: {x}^{2} - {y}^{2} = (x + y)(x - y)$

$\bf \: {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)$

$\bf \: {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y)$

$\bf \: {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - yx + {y}^{2} )$

$\bf \: {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2} )$