Factor 4a^8-53a^4b^4+49b^8

$4 a^{8} - 53 a^{4} b^{4} + 49 b^{8}$

Rewrite the middle term.

$4 a^{8} + 2 (2 a^{4}) (7 b^{4}) - 81 a^{4} b^{4} + 49 b^{8}$

Rearrange terms.

$4 a^{8} + 2 (2 a^{4}) (7 b^{4}) + 49 b^{8} - 81 a^{4} b^{4}$

Factor first three terms by perfect square rule.

${(2 a^{4} + 7 b^{4})}^{2} - 81 a^{4} b^{4}$

Rewrite $81 a^{4} b^{4}$ as ${(9 a^{2} b^{2})}^{2}$ .

${(2 a^{4} + 7 b^{4})}^{2} - {(9 a^{2} b^{2})}^{2}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 a^{4} + 7 b^{4}$ and $b = 9 a^{2} b^{2}$ .

$(2 a^{4} + 7 b^{4} + 9 a^{2} b^{2}) (2 a^{4} + 7 b^{4} - (9 a^{2} b^{2}))$

Simplify.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 7 = 14$ and whose sum is $b = 9$ .

Reorder $7 b^{4}$ and $9 a^{2} b^{2}$ .

$(2 a^{4} + 9 a^{2} b^{2} + 7 b^{4}) (2 a^{4} + 7 b^{4} - (9 a^{2} b^{2}))$

Factor $9$ out of $9 a^{2} b^{2}$ .

$(2 a^{4} + 9 (a^{2} b^{2}) + 7 b^{4}) (2 a^{4} + 7 b^{4} - (9 a^{2} b^{2}))$

Rewrite $9$ as $2$ plus $7$

$(2 a^{4} + (2 + 7) (a^{2} b^{2}) + 7 b^{4}) (2 a^{4} + 7 b^{4} - (9 a^{2} b^{2}))$

Apply the distributive property.

$(2 a^{4} + 2 (a^{2} b^{2}) + 7 (a^{2} b^{2}) + 7 b^{4}) (2 a^{4} + 7 b^{4} - (9 a^{2} b^{2}))$

Remove unnecessary parentheses.

$(2 a^{4} + 2 a^{2} b^{2} + 7 (a^{2} b^{2}) + 7 b^{4}) (2 a^{4} + 7 b^{4} - (9 a^{2} b^{2}))$

Remove unnecessary parentheses.

$(2 a^{4} + 2 a^{2} b^{2} + 7 a^{2} b^{2} + 7 b^{4}) (2 a^{4} + 7 b^{4} - (9 a^{2} b^{2}))$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$((2 a^{4} + 2 a^{2} b^{2}) + 7 a^{2} b^{2} + 7 b^{4}) (2 a^{4} + 7 b^{4} - (9 a^{2} b^{2}))$

Factor out the greatest common factor (GCF) from each group.

$(2 a^{2} (a^{2} + b^{2}) + 7 b^{2} (a^{2} + b^{2})) (2 a^{4} + 7 b^{4} - (9 a^{2} b^{2}))$

Factor the polynomial by factoring out the greatest common factor, $a^{2} + b^{2}$ .

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (2 a^{4} + 7 b^{4} - (9 a^{2} b^{2}))$

Multiply $9$ by $- 1$ .

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (2 a^{4} + 7 b^{4} - 9 (a^{2} b^{2}))$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 7 = 14$ and whose sum is $b = - 9$ .

Reorder terms.

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (2 a^{4} + 7 b^{4} - 9 a^{2} b^{2})$

Reorder $7 b^{4}$ and $- 9 a^{2} b^{2}$ .

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (2 a^{4} - 9 a^{2} b^{2} + 7 b^{4})$

Factor $- 9$ out of $- 9 a^{2} b^{2}$ .

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (2 a^{4} - 9 (a^{2} b^{2}) + 7 b^{4})$

Rewrite $- 9$ as $- 2$ plus $- 7$

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (2 a^{4} + (- 2 - 7) (a^{2} b^{2}) + 7 b^{4})$

Apply the distributive property.

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (2 a^{4} - 2 (a^{2} b^{2}) - 7 (a^{2} b^{2}) + 7 b^{4})$

Remove unnecessary parentheses.

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (2 a^{4} - 2 a^{2} b^{2} - 7 (a^{2} b^{2}) + 7 b^{4})$

Remove unnecessary parentheses.

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (2 a^{4} - 2 a^{2} b^{2} - 7 a^{2} b^{2} + 7 b^{4})$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) ((2 a^{4} - 2 a^{2} b^{2}) - 7 a^{2} b^{2} + 7 b^{4})$

Factor out the greatest common factor (GCF) from each group.

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (2 a^{2} (a^{2} - b^{2}) - 7 b^{2} (a^{2} - b^{2}))$

Factor the polynomial by factoring out the greatest common factor, $a^{2} - b^{2}$ .

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) ((a^{2} - b^{2}) (2 a^{2} - 7 b^{2}))$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = b$ .

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) ((a + b) (a - b) (2 a^{2} - 7 b^{2}))$

Remove unnecessary parentheses.

$(a^{2} + b^{2}) (2 a^{2} + 7 b^{2}) (a + b) (a - b) (2 a^{2} - 7 b^{2})$

Factor 4a^8-53a^4b^4+49b^8

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