Answer :
We start with the polynomial
$$
x^3 - 5x^2 - 9x + 45.
$$
**Step 1. Identify a Root**
A good strategy for factoring a cubic polynomial is to look for an integer root by testing simple values. By substituting $x = 5$, we see:
$$
5^3 - 5(5)^2 - 9(5) + 45 = 125 - 125 - 45 + 45 = 0.
$$
Since the polynomial equals zero, $x = 5$ is a root. This means that $(x - 5)$ is a factor of the polynomial.
**Step 2. Factor the Polynomial by Division**
To factor the polynomial further, divide it by $(x - 5)$. This division yields a quadratic polynomial:
$$
x^3 - 5x^2 - 9x + 45 = (x - 5)(x^2 - 9).
$$
**Step 3. Factor the Quadratic**
Notice that the quadratic
$$
x^2 - 9
$$
is a difference of squares. Recall that a difference of squares factors as
$$
a^2 - b^2 = (a - b)(a + b).
$$
Here, $a = x$ and $b = 3$, so
$$
x^2 - 9 = (x - 3)(x + 3).
$$
Thus, the complete factorization of the polynomial is:
$$
x^3 - 5x^2 - 9x + 45 = (x - 5)(x - 3)(x + 3).
$$
**Step 4. Find the Roots**
To find the roots of the polynomial, set each factor equal to zero:
1. From $x - 5 = 0$, we have $x = 5$.
2. From $x - 3 = 0$, we have $x = 3$.
3. From $x + 3 = 0$, we have $x = -3$.
The solutions (or roots) of the equation are:
$$
x = -3,\quad x = 3,\quad x = 5.
$$
**Final Answer**
The factorization of the polynomial is
$$
(x - 5)(x - 3)(x + 3),
$$
and the roots are $x = -3$, $3$, and $5$.
$$
x^3 - 5x^2 - 9x + 45.
$$
**Step 1. Identify a Root**
A good strategy for factoring a cubic polynomial is to look for an integer root by testing simple values. By substituting $x = 5$, we see:
$$
5^3 - 5(5)^2 - 9(5) + 45 = 125 - 125 - 45 + 45 = 0.
$$
Since the polynomial equals zero, $x = 5$ is a root. This means that $(x - 5)$ is a factor of the polynomial.
**Step 2. Factor the Polynomial by Division**
To factor the polynomial further, divide it by $(x - 5)$. This division yields a quadratic polynomial:
$$
x^3 - 5x^2 - 9x + 45 = (x - 5)(x^2 - 9).
$$
**Step 3. Factor the Quadratic**
Notice that the quadratic
$$
x^2 - 9
$$
is a difference of squares. Recall that a difference of squares factors as
$$
a^2 - b^2 = (a - b)(a + b).
$$
Here, $a = x$ and $b = 3$, so
$$
x^2 - 9 = (x - 3)(x + 3).
$$
Thus, the complete factorization of the polynomial is:
$$
x^3 - 5x^2 - 9x + 45 = (x - 5)(x - 3)(x + 3).
$$
**Step 4. Find the Roots**
To find the roots of the polynomial, set each factor equal to zero:
1. From $x - 5 = 0$, we have $x = 5$.
2. From $x - 3 = 0$, we have $x = 3$.
3. From $x + 3 = 0$, we have $x = -3$.
The solutions (or roots) of the equation are:
$$
x = -3,\quad x = 3,\quad x = 5.
$$
**Final Answer**
The factorization of the polynomial is
$$
(x - 5)(x - 3)(x + 3),
$$
and the roots are $x = -3$, $3$, and $5$.
