Calculus Examples

$5 x^{4} - 24 x^{2} + 16$

Step 1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$5 {(x^{2})}^{2} - 24 x^{2} + 16$

Step 2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$5 u^{2} - 24 u + 16$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 5 \cdot 16 = 80$ and whose sum is $b = - 24$ .

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Step 3.1.1

Factor $- 24$ out of $- 24 u$ .

$5 u^{2} - 24 (u) + 16$

Step 3.1.2

Rewrite $- 24$ as $- 4$ plus $- 20$

$5 u^{2} + (- 4 - 20) u + 16$

Step 3.1.3

Apply the distributive property.

$5 u^{2} - 4 u - 20 u + 16$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(5 u^{2} - 4 u) - 20 u + 16$

Step 3.2.2

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

$u (5 u - 4) - 4 (5 u - 4)$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $5 u - 4$ .

$(5 u - 4) (u - 4)$

Step 4

Replace all occurrences of $u$ with $x^{2}$ .

$(5 x^{2} - 4) (x^{2} - 4)$

Step 5

Rewrite $4$ as $2^{2}$ .

$(5 x^{2} - 4) (x^{2} - 2^{2})$

Step 6

Factor.

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Step 6.1

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

$(5 x^{2} - 4) ((x + 2) (x - 2))$

Step 6.2

Remove unnecessary parentheses.

$(5 x^{2} - 4) (x + 2) (x - 2)$