Question

If y-z = 4, yz = -5, then find the value of y3 – z3.

Answer 1

Answer:

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Answer 2

If $y - z = 4$ and $yz = -5$, then the value of $y^{3}- z^{3}$ is equal to 4.

Step-by-step Explanation

Given: $y-z=4$ and $yz=-5$

To be found:

To find the value of $y^{3}- z^{3}$

Solution:

Here, let's first recall the identity,

$y^{3}- z^{3}=(y-z)(y^{2}+yz+z^{2} )$ -------------(1)

Now, substituting the given values in (1), we get

$y^{3}- z^{3}=(4)(y^{2}+(-5)+z^{2} )\\\implies y^{3}- z^{3}=4(y^{2}+z^{2}-5 )$ -------------(2)

Also, we are familiar with identity,

$(y-z)^{2} =y^{2} +z^{2}-2yz$  -------------(3)

Substituting the given values in (3), we get

$(4)^{2} =y^{2} +z^{2} -2(-5)\\\implies16=y^{2} +z^{2} +10\\\implies 16-10=y^{2} +z^{2}$

Rearranging, we get

$y^{2} +z^{2} =6$  --------(4)

Now substituting (4) in (2), we get

$y^{3}- z^{3}=4(6-5)\\\implies y^{3}- z^{3}=4 \times 1\\\implies y^{3}- z^{3}=4$

Therefore, the value of $y^{3}- z^{3}$ when $y-z=4$ and $yz=-5$ is 4.

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