Simplify (k^14+4k^7+3)/(2k^11+2k^4)

$\frac{k^{14} + 4 k^{7} + 3}{2 k^{11} + 2 k^{4}}$

Step 1

Simplify the numerator.

Rewrite $k^{14}$ as ${(k^{7})}^{2}$ .

$\frac{{(k^{7})}^{2} + 4 k^{7} + 3}{2 k^{11} + 2 k^{4}}$

Let $u = k^{7}$ . Substitute $u$ for all occurrences of $k^{7}$ .

$\frac{u^{2} + 4 u + 3}{2 k^{11} + 2 k^{4}}$

Factor $u^{2} + 4 u + 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Write the factored form using these integers.

$\frac{(u + 1) (u + 3)}{2 k^{11} + 2 k^{4}}$

Replace all occurrences of $u$ with $k^{7}$ .

$\frac{(k^{7} + 1) (k^{7} + 3)}{2 k^{11} + 2 k^{4}}$

Step 2

Simplify terms.

Factor $2 k^{4}$ out of $2 k^{11} + 2 k^{4}$ .

Factor $2 k^{4}$ out of $2 k^{11}$ .

$\frac{(k^{7} + 1) (k^{7} + 3)}{2 k^{4} (k^{7}) + 2 k^{4}}$

Factor $2 k^{4}$ out of $2 k^{4}$ .

$\frac{(k^{7} + 1) (k^{7} + 3)}{2 k^{4} (k^{7}) + 2 k^{4} (1)}$

Factor $2 k^{4}$ out of $2 k^{4} (k^{7}) + 2 k^{4} (1)$ .

$\frac{(k^{7} + 1) (k^{7} + 3)}{2 k^{4} (k^{7} + 1)}$

Cancel the common factor of $k^{7} + 1$ .

Cancel the common factor.

$\frac{(k^{7} + 1) (k^{7} + 3)}{2 k^{4} (k^{7} + 1)}$

Rewrite the expression.

$\frac{k^{7} + 3}{2 k^{4}}$

Simplify (k^14+4k^7+3)/(2k^11+2k^4)

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