Write as a Set of Linear Factors f(x)=2x^3+7x^2+98x+343

$f (x) = 2 x^{3} + 7 x^{2} + 98 x + 343$

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = (2 x^{3} + 7 x^{2}) + 98 x + 343$

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

$f (x) = x^{2} (2 x + 7) + 49 (2 x + 7)$

Step 2

Factor the polynomial by factoring out the greatest common factor, $2 x + 7$ .

$f (x) = (2 x + 7) (x^{2} + 49)$

Step 3

Factor over the complex numbers.

Multiply the constant in the polynomial $x^{2} + 49$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$f (x) = (2 x + 7) (x^{2} - 49 i^{2})$

Rewrite $49 i^{2}$ as ${(7 i)}^{2}$ .

$f (x) = (2 x + 7) (x^{2} - {(7 i)}^{2})$

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

$f (x) = (2 x + 7) ((x + 7 i) (x - (7 i)))$

Factor.

Multiply $7$ by $- 1$ .

$f (x) = (2 x + 7) ((x + 7 i) (x - 7 i))$

Remove unnecessary parentheses.

$f (x) = (2 x + 7) (x + 7 i) (x - 7 i)$