Fictorize fully 1. $$ t^{2}(r-3)-r+5 $$ 9. $$ \frac{x^{3}+1}{x^{2}-x+1} $$

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Certainly! Let's fully factorize the given expressions step by step.

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## Problem 1: Factorize $$ t^{2}(r - 3) - r + 5 $$

### Step 1: Expand the Expression
First, distribute $$ t^{2} $$ across $$ (r - 3) $$:

$$
t^{2}(r - 3) - r + 5 = r t^{2} - 3 t^{2} - r + 5
$$

### Step 2: Rearrange the Terms
Group like terms to make factoring easier:

$$
r t^{2} - r - 3 t^{2} + 5 = r(t^{2} - 1) - 3 t^{2} + 5
$$

### Step 3: Factor Common Terms
Notice that $$ t^{2} - 1 $$ is a difference of squares and can be factored further:

$$
t^{2} - 1 = (t - 1)(t + 1)
$$

So, substitute back into the expression:

$$
r(t - 1)(t + 1) - 3 t^{2} + 5
$$

### Step 4: Assess Further Factorization
At this point, the expression:

$$
r(t - 1)(t + 1) - 3 t^{2} + 5
$$

does not have any common factors that can be factored out further with integer coefficients. Therefore, the fully factorized form of the original expression is:

$$
r(t - 1)(t + 1) - 3 t^{2} + 5
$$

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## Problem 9: Factorize $$ \frac{x^{3} + 1}{x^{2} - x + 1} $$

### Step 1: Factorize the Numerator
The numerator $$ x^{3} + 1 $$ is a sum of cubes, which can be factored using the formula:

$$
a^{3} + b^{3} = (a + b)(a^{2} - a b + b^{2})
$$

Let $$ a = x $$ and $$ b = 1 $$:

$$
x^{3} + 1 = (x + 1)(x^{2} - x \cdot 1 + 1^{2}) = (x + 1)(x^{2} - x + 1)
$$

### Step 2: Simplify the Expression
Now, substitute the factored form of the numerator back into the original expression:

$$
\frac{x^{3} + 1}{x^{2} - x + 1} = \frac{(x + 1)(x^{2} - x + 1)}{x^{2} - x + 1}
$$

Notice that $$ x^{2} - x + 1 $$ appears in both the numerator and the denominator, so they cancel each other out:

$$
\frac{(x + 1)\cancel{(x^{2} - x + 1)}}{\cancel{x^{2} - x + 1}} = x + 1
$$

### Step 3: Final Simplified Form
After cancellation, the simplified (and fully factorized) form of the expression is:

$$
x + 1
$$

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**Summary:**

1. **Problem 1:** The expression $$ t^{2}(r - 3) - r + 5 $$ is fully factorized as:

$$
   r(t - 1)(t + 1) - 3 t^{2} + 5
   $$

2. **Problem 9:** The expression $$ \frac{x^{3} + 1}{x^{2} - x + 1} $$ simplifies and factorizes to:

$$
   x + 1
   $$

If you have any further questions or need additional assistance with these problems, feel free to ask!
Problem 1: $$ t^{2}(r - 3) - r + 5 $$ factors to $$ r(t - 1)(t + 1) - 3t^{2} + 5 $$. Problem 9: $$ \frac{x^{3} + 1}{x^{2} - x + 1} $$ simplifies to $$ x + 1 $$.

Fictorize fully 1. \( t^{2}(r-3)-r+5 \) 9. \( \frac{x^{3}+1}{x^{2}-x+1} \)

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