find all zeros of P(x)= -3x^3+5x^2+19x-21

To find the zeros of the polynomial \( P(x) = -3x^3 + 5x^2 + 19x - 21 \), we must solve for \( x \) where \( P(x) = 0 \).

### Step 1: Use the Rational Root Theorem
This theorem helps us find potential rational roots. For a polynomial \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0 \), the potential rational roots are of the form:

\[
\frac{\text{factors of } a_0}{\text{factors of } a_n}
\]

Here, \( a_n = -3 \) (leading coefficient) and \( a_0 = -21 \) (constant term).

Factors of \( -21 \): \( \pm 1, \pm 3, \pm 7, \pm 21 \)

Factors of \( -3 \): \( \pm 1, \pm 3 \)

Potential rational roots: 
\[
\pm 1, \pm \frac{1}{3}, \pm 3, \pm 7, \pm \frac{7}{3}, \pm 21
\]

### Step 2: Test potential roots using synthetic division or direct substitution

Let’s check \( x = 1 \):

\[
P(1) = -3(1)^3 + 5(1)^2 + 19(1) - 21 = -3 + 5 + 19 - 21 = 0
\]

So, \( x = 1 \) is a root.

### Step 3: Polynomial Division
Now, we divide \( P(x) \) by \( (x - 1) \) using synthetic or long division.

After dividing \( -3x^3 + 5x^2 + 19x - 21 \) by \( x - 1 \), the quotient is:

\[
-3x^2 + 2x + 21
\]

### Step 4: Solve the quadratic equation
Now, solve \( -3x^2 + 2x + 21 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For \( -3x^2 + 2x + 21 \), we have \( a = -3 \), \( b = 2 \), and \( c = 21 \).

\[
x = \frac{-2 \pm \sqrt{(2)^2 - 4(-3)(21)}}{2(-3)} = \frac{-2 \pm \sqrt{4 + 252}}{-6} = \frac{-2 \pm \sqrt{256}}{-6}
\]

\[
x = \frac{-2 \pm 16}{-6}
\]

Thus, the solutions are:

\[
x = \frac{-2 + 16}{-6} = \frac{14}{-6} = -\frac{7}{3}
\]
\[
x = \frac{-2 - 16}{-6} = \frac{-18}{-6} = 3
\]

### Step 5: List the solutions
The zeros of the polynomial are:
\[
x = 1, \quad x = -\frac{7}{3}, \quad x = 3
\]

Would you like a detailed explanation for any part of this process?

Here are 5 related questions:
1. How can we use synthetic division to simplify polynomial division?
2. What are the steps for applying the Rational Root Theorem?
3. How can we solve a quadratic equation using factoring instead of the quadratic formula?
4. What is the significance of complex roots in polynomials?
5. How can we check if all solutions of a polynomial are correct?

**Tip:** Always test possible rational roots first before proceeding to complex or irrational roots. This simplifies solving cubic and higher-degree polynomials.

Solve for the zeros of the cubic polynomial -3x^3 + 5x^2 + 19x - 21 using the Rational Root Theorem and the quadratic formula. Follow this detailed solution to understand key algebraic concepts, perfect for students in Grades 10-12.

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