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We are given the polynomial \( t(x) = x^3 - 5x^2 - 9x + 45 \) and asked to show that \( x - 5 \) is a factor of the polynomial. Then we need to factor the polynomial completely.

### Step 1: Verify if \( x - 5 \) is a factor using the Factor Theorem
According to the Factor Theorem, \( x - 5 \) is a factor of \( t(x) \) if and only if \( t(5) = 0 \).

Let's substitute \( x = 5 \) into \( t(x) \):

\[
t(5) = 5^3 - 5(5^2) - 9(5) + 45
\]
\[
t(5) = 125 - 5(25) - 45 + 45
\]
\[
t(5) = 125 - 125 - 45 + 45
\]
\[
t(5) = 0
\]

Since \( t(5) = 0 \), \( x - 5 \) is indeed a factor of the polynomial.

### Step 2: Perform synthetic division to factor \( t(x) \)

Now, we divide \( t(x) = x^3 - 5x^2 - 9x + 45 \) by \( x - 5 \) using synthetic division:

1. Write down the coefficients of \( t(x) \): \( 1, -5, -9, 45 \).
2. Perform synthetic division using \( x = 5 \):

\[
\begin{array}{r|rrrr}
5 & 1 & -5 & -9 & 45 \\
  &   & 5  & 0  & -45 \\
\hline
  & 1 & 0  & -9 & 0 \\
\end{array}
\]

The quotient is \( x^2 - 9 \), and the remainder is 0, confirming that the division is exact.

### Step 3: Factor \( x^2 - 9 \)

The quotient \( x^2 - 9 \) is a difference of squares, which can be factored as:

\[
x^2 - 9 = (x - 3)(x + 3)
\]

### Step 4: Final factorization

Thus, the complete factorization of \( t(x) \) is:

\[
t(x) = (x - 5)(x - 3)(x + 3)
\]

### Final Answer:
The polynomial \( t(x) = x^3 - 5x^2 - 9x + 45 \) factors completely as:

\[
t(x) = (x - 5)(x - 3)(x + 3)
\]

Would you like more details or have any questions? Here are five related questions to further explore this topic:

1. How can the Factor Theorem be applied to determine if a binomial is a factor?
2. What are some other methods for factoring cubic polynomials?
3. Can synthetic division be used if the divisor is more complex than \( x - a \)?
4. What is the general formula for factoring a difference of squares?
5. How does the remainder theorem relate to the factor theorem?

**Tip:** When factoring polynomials, always check for special forms like difference of squares and perfect square trinomials to simplify the process.

Show that the binomial x - 5 is a factor of the polynomial t(x) = x^3 - 5x^2 - 9x + 45. Then factor the polynomial completely.

Learn how to show that x - 5 is a factor of the polynomial x^3 - 5x^2 - 9x + 45 using the Factor Theorem. Complete solution includes synthetic division and factoring the polynomial into its final form: (x - 5)(x - 3)(x + 3). Suitable for Grades 10-12.

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