Write as a Set of Linear Factors P(x)=x^4+3x^3+49x^2+147x

$P (x) = x^{4} + 3 x^{3} + 49 x^{2} + 147 x$

Step 1

Factor $x$ out of $x^{4} + 3 x^{3} + 49 x^{2} + 147 x$ .

Factor $x$ out of $x^{4}$ .

$P (x) = x \cdot x^{3} + 3 x^{3} + 49 x^{2} + 147 x$

Factor $x$ out of $3 x^{3}$ .

$P (x) = x \cdot x^{3} + x (3 x^{2}) + 49 x^{2} + 147 x$

Factor $x$ out of $49 x^{2}$ .

$P (x) = x \cdot x^{3} + x (3 x^{2}) + x (49 x) + 147 x$

Factor $x$ out of $147 x$ .

$P (x) = x \cdot x^{3} + x (3 x^{2}) + x (49 x) + x \cdot 147$

Factor $x$ out of $x \cdot x^{3} + x (3 x^{2})$ .

$P (x) = x (x^{3} + 3 x^{2}) + x (49 x) + x \cdot 147$

Factor $x$ out of $x (x^{3} + 3 x^{2}) + x (49 x)$ .

$P (x) = x (x^{3} + 3 x^{2} + 49 x) + x \cdot 147$

Factor $x$ out of $x (x^{3} + 3 x^{2} + 49 x) + x \cdot 147$ .

$P (x) = x (x^{3} + 3 x^{2} + 49 x + 147)$

Step 2

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$P (x) = x ((x^{3} + 3 x^{2}) + 49 x + 147)$

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

$P (x) = x (x^{2} (x + 3) + 49 (x + 3))$

Step 3

Factor.

Factor the polynomial by factoring out the greatest common factor, $x + 3$ .

$P (x) = x ((x + 3) (x^{2} + 49))$

Remove unnecessary parentheses.

$P (x) = x (x + 3) (x^{2} + 49)$

Step 4

Factor over the complex numbers.

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

$P (x) = x (x + 3) (x^{2} - 49 i^{2})$

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

$P (x) = x (x + 3) (x^{2} - {(7 i)}^{2})$

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 $i = 7 i$ .

$P (x) = x (x + 3) ((x + 7 i) (x - (7 i)))$

Factor.

Multiply $7$ by $- 1$ .

$P (x) = x (x + 3) ((x + 7 i) (x - 7 i))$

Remove unnecessary parentheses.

$P (x) = x (x + 3) (x + 7 i) (x - 7 i)$