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Factors of a³ + b³

Simplify the ...

Question

Simplify the following expression:
$(5 x - 7 y)^{3} + (5 x + 7 y)^{3}$
(4 Marks)

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Solution

Given expression: $(5 x - 7 y)^{3} + (5 x + 7 y)^{3}$
It is in the form of $a^{3} + b^{3}$
We know that, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$

$(5 x - 7 y)^{3} + (5 x + 7 y)^{3}$ $= [(5 x - 7 y) + (5 x + 7 y)] \times = [{(5 x - 7 y)}^{2} - (5 x - 7 y) \times (5 x + 7 y) + {(5 x + 7 y)}^{2}]$
$= (10 x) \times [{(5 x - 7 y)}^{2} - (5 x - 7 y) (5 x + 7 y) + {(5 x + 7 y)}^{2}]$
-------- eq. (1)
(1 Mark)

Now,
${(5 x - 7 y)}^{2}$
It is in the form of $(a - b)^{2}$
We know that,
$(a - b)^{2} = a^{2} + b^{2} - 2 a b \Rightarrow {(5 x - 7 y)}^{2} = (5 x)^{2} + {(7 y)}^{2} - 2 \times 5 x \times 7 y = 25 x^{2} + 49 y^{2} - 70 x y$

Now,
$(5 x - 7 y) (5 x + 7 y)$
It is in the form of $(a - b) (a + b)$
We know that,
$(a - b) (a + b) = a^{2} - b^{2} \Rightarrow {(5 x)}^{2} - {(7 y)}^{2} = 25 x^{2} - 49 y^{2}$

Now,
${(5 x + 7 y)}^{2}$
It is in the form of $(a + b)^{2}$
We know that,
$(a + b)^{2} = a^{2} + b^{2} + 2 a b \Rightarrow {(5 x + 7 y)}^{2} = (5 x)^{2} + {(7 y)}^{2} + 2 \times 5 x \times 7 y = 25 x^{2} + 49 y^{2} + 70 x y$
(2 Marks)

By substituting, $(5 x - 7 y)^{2}$ ; $(5 x - 7 y) (5 x + 7 y)$ and $(5 x + 7 y)^{2}$ in eq. (1); we get,
$= (10 x) \times [(25 x^{2} + 49 y^{2} - 70 x y) - (25 x^{2} - 49 y^{2}) + (25 x^{2} + 49 y^{2} + 70 x y)]$

$= (10 x) [25 x^{2} + 147 y^{2}] = 250 x^{3} + 1470 x y^{2}$
(1 Mark)

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