Answer :
We begin with the expression
[tex]$$
x^3 - 9x + 5x^2 - 45.
$$[/tex]
Step 1. Rearranging the terms
First, we rearrange the terms in descending order of the power of [tex]$x$[/tex]:
[tex]$$
x^3 + 5x^2 - 9x - 45.
$$[/tex]
Step 2. Factor by grouping
Next, we group the terms in a way that common factors can be factored out:
[tex]$$
\begin{aligned}
x^3 + 5x^2 - 9x - 45 &= \left(x^3 + 5x^2\right) + \left(-9x - 45\right).
\end{aligned}
$$[/tex]
Factor out the greatest common factor (GCF) from each group:
- In the first group, factor out [tex]$x^2$[/tex]:
[tex]$$
x^3 + 5x^2 = x^2(x + 5).
$$[/tex]
- In the second group, factor out [tex]$-9$[/tex]:
[tex]$$
-9x - 45 = -9(x + 5).
$$[/tex]
Now, the expression becomes:
[tex]$$
x^2(x + 5) - 9(x + 5).
$$[/tex]
Step 3. Factor out the common binomial
Since both terms contain the factor [tex]$(x + 5)$[/tex], we factor it out:
[tex]$$
(x + 5)(x^2 - 9).
$$[/tex]
Step 4. Factor the difference of squares
Notice that [tex]$x^2 - 9$[/tex] is a difference of squares, which can be factored as:
[tex]$$
x^2 - 9 = (x - 3)(x + 3).
$$[/tex]
Thus, the complete factorization of the polynomial is:
[tex]$$
(x + 5)(x - 3)(x + 3).
$$[/tex]
It is common to write the factors in a different order. For example, rearranging them gives:
[tex]$$
(x - 3)(x + 3)(x + 5).
$$[/tex]
Step 5. Finding the roots
The roots of the polynomial are the solutions to the equation:
[tex]$$
(x - 3)(x + 3)(x + 5) = 0.
$$[/tex]
Setting each factor equal to zero gives:
1. [tex]$x - 3 = 0 \quad \Rightarrow \quad x = 3,$[/tex]
2. [tex]$x + 3 = 0 \quad \Rightarrow \quad x = -3,$[/tex]
3. [tex]$x + 5 = 0 \quad \Rightarrow \quad x = -5.$[/tex]
Final Answer
- The polynomial in standard form is:
[tex]$$
x^3 + 5x^2 - 9x - 45.
$$[/tex]
- Its factorization is:
[tex]$$
(x - 3)(x + 3)(x + 5).
$$[/tex]
- The roots of the polynomial are:
[tex]$$
x = 3,\quad x = -3,\quad x = -5.
$$[/tex]
[tex]$$
x^3 - 9x + 5x^2 - 45.
$$[/tex]
Step 1. Rearranging the terms
First, we rearrange the terms in descending order of the power of [tex]$x$[/tex]:
[tex]$$
x^3 + 5x^2 - 9x - 45.
$$[/tex]
Step 2. Factor by grouping
Next, we group the terms in a way that common factors can be factored out:
[tex]$$
\begin{aligned}
x^3 + 5x^2 - 9x - 45 &= \left(x^3 + 5x^2\right) + \left(-9x - 45\right).
\end{aligned}
$$[/tex]
Factor out the greatest common factor (GCF) from each group:
- In the first group, factor out [tex]$x^2$[/tex]:
[tex]$$
x^3 + 5x^2 = x^2(x + 5).
$$[/tex]
- In the second group, factor out [tex]$-9$[/tex]:
[tex]$$
-9x - 45 = -9(x + 5).
$$[/tex]
Now, the expression becomes:
[tex]$$
x^2(x + 5) - 9(x + 5).
$$[/tex]
Step 3. Factor out the common binomial
Since both terms contain the factor [tex]$(x + 5)$[/tex], we factor it out:
[tex]$$
(x + 5)(x^2 - 9).
$$[/tex]
Step 4. Factor the difference of squares
Notice that [tex]$x^2 - 9$[/tex] is a difference of squares, which can be factored as:
[tex]$$
x^2 - 9 = (x - 3)(x + 3).
$$[/tex]
Thus, the complete factorization of the polynomial is:
[tex]$$
(x + 5)(x - 3)(x + 3).
$$[/tex]
It is common to write the factors in a different order. For example, rearranging them gives:
[tex]$$
(x - 3)(x + 3)(x + 5).
$$[/tex]
Step 5. Finding the roots
The roots of the polynomial are the solutions to the equation:
[tex]$$
(x - 3)(x + 3)(x + 5) = 0.
$$[/tex]
Setting each factor equal to zero gives:
1. [tex]$x - 3 = 0 \quad \Rightarrow \quad x = 3,$[/tex]
2. [tex]$x + 3 = 0 \quad \Rightarrow \quad x = -3,$[/tex]
3. [tex]$x + 5 = 0 \quad \Rightarrow \quad x = -5.$[/tex]
Final Answer
- The polynomial in standard form is:
[tex]$$
x^3 + 5x^2 - 9x - 45.
$$[/tex]
- Its factorization is:
[tex]$$
(x - 3)(x + 3)(x + 5).
$$[/tex]
- The roots of the polynomial are:
[tex]$$
x = 3,\quad x = -3,\quad x = -5.
$$[/tex]
