$$ 3 x^{2}+25 x-18 $$ Which of the following is a factor of the polynomial above? $$ x-9 $$ $$ 3 x+3 $$ $$ 3 x-2 $$ $$ 3 x+1 $$

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To determine which of the given options is a factor of the polynomial $$ 3x^{2} + 25x - 18 $$, we can use the factor theorem. According to the factor theorem, if $$ x - c $$ is a factor of a polynomial $$ P(x) $$, then $$ P(c) = 0 $$.

We will evaluate the polynomial at the roots of the factors provided:

1. For $$ x - 9 $$, we will evaluate $$ P(9) $$.
2. For $$ 3x + 3 $$, we will evaluate $$ P(-1) $$ (since $$ 3x + 3 = 0 $$ gives $$ x = -1 $$).
3. For $$ 3x - 2 $$, we will evaluate $$ P(\frac{2}{3}) $$.
4. For $$ 3x + 1 $$, we will evaluate $$ P(-\frac{1}{3}) $$.

Let's calculate these values step by step.

### Step 1: Evaluate $$ P(9) $$
$$
P(9) = 3(9)^{2} + 25(9) - 18
$$

### Step 2: Evaluate $$ P(-1) $$
$$
P(-1) = 3(-1)^{2} + 25(-1) - 18
$$

### Step 3: Evaluate $$ P(\frac{2}{3}) $$
$$
P\left(\frac{2}{3}\right) = 3\left(\frac{2}{3}\right)^{2} + 25\left(\frac{2}{3}\right) - 18
$$

### Step 4: Evaluate $$ P(-\frac{1}{3}) $$
$$
P\left(-\frac{1}{3}\right) = 3\left(-\frac{1}{3}\right)^{2} + 25\left(-\frac{1}{3}\right) - 18
$$

Now, let's perform these calculations.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$3x^{2}+25x-18$$
- step1: Substitute:
  $$3\left(-1\right)^{2}+25\left(-1\right)-18$$
- step2: Evaluate the power:
  $$3\times 1+25\left(-1\right)-18$$
- step3: Multiply:
  $$3+25\left(-1\right)-18$$
- step4: Simplify:
  $$3-25-18$$
- step5: Subtract the numbers:
  $$-40$$
Substitute $$ x=-1/3 $$ into the expression $$ 3x^2 + 25x - 18 $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$3x^{2}+25x-18$$
- step1: Substitute:
  $$3\left(\frac{-1}{3}\right)^{2}+25\times \frac{-1}{3}-18$$
- step2: Rewrite the fraction:
  $$3\left(-\frac{1}{3}\right)^{2}+25\times \frac{-1}{3}-18$$
- step3: Rewrite the fraction:
  $$3\left(-\frac{1}{3}\right)^{2}+25\left(-\frac{1}{3}\right)-18$$
- step4: Multiply the terms:
  $$\frac{1}{3}+25\left(-\frac{1}{3}\right)-18$$
- step5: Multiply the numbers:
  $$\frac{1}{3}-\frac{25}{3}-18$$
- step6: Reduce fractions to a common denominator:
  $$\frac{1}{3}-\frac{25}{3}-\frac{18\times 3}{3}$$
- step7: Transform the expression:
  $$\frac{1-25-18\times 3}{3}$$
- step8: Multiply the numbers:
  $$\frac{1-25-54}{3}$$
- step9: Subtract the numbers:
  $$\frac{-78}{3}$$
- step10: Reduce the numbers:
  $$\frac{-26}{1}$$
- step11: Calculate:
  $$-26$$
Substitute $$ x=9 $$ into the expression $$ 3x^2 + 25x - 18 $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$3x^{2}+25x-18$$
- step1: Substitute:
  $$3\times 9^{2}+25\times 9-18$$
- step2: Multiply the terms:
  $$243+25\times 9-18$$
- step3: Multiply the numbers:
  $$243+225-18$$
- step4: Calculate:
  $$450$$
Substitute $$ x=2/3 $$ into the expression $$ 3x^2 + 25x - 18 $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$3x^{2}+25x-18$$
- step1: Substitute:
  $$3\left(\frac{2}{3}\right)^{2}+25\times \frac{2}{3}-18$$
- step2: Multiply the terms:
  $$\frac{4}{3}+25\times \frac{2}{3}-18$$
- step3: Multiply the numbers:
  $$\frac{4}{3}+\frac{50}{3}-18$$
- step4: Reduce fractions to a common denominator:
  $$\frac{4}{3}+\frac{50}{3}-\frac{18\times 3}{3}$$
- step5: Transform the expression:
  $$\frac{4+50-18\times 3}{3}$$
- step6: Multiply the numbers:
  $$\frac{4+50-54}{3}$$
- step7: Calculate:
  $$\frac{0}{3}$$
- step8: Calculate:
  $$0$$
Here are the results of the evaluations:

1. $$ P(9) = 450 $$
2. $$ P(-1) = -40 $$
3. $$ P\left(\frac{2}{3}\right) = 0 $$
4. $$ P\left(-\frac{1}{3}\right) = -26 $$

### Conclusion:
- Since $$ P\left(\frac{2}{3}\right) = 0 $$, this means that $$ 3x - 2 $$ is a factor of the polynomial $$ 3x^{2} + 25x - 18 $$.

Thus, the factor of the polynomial is $$ \boxed{3x - 2} $$.
$$ 3x - 2 $$ is a factor of the polynomial $$ 3x^{2} + 25x - 18 $$.

\( 3 x^{2}+25 x-18 \) Which of the following is a factor of the polynomial above? \( x-9 \) \( 3 x+3 \) \( 3 x-2 \) \( 3 x+1 \)

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