Math Problem Statement

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

Solution

To find the zeros of the polynomial P(x)=3x3+5x2+19x21P(x) = -3x^3 + 5x^2 + 19x - 21, we must solve for xx where P(x)=0P(x) = 0.

Step 1: Use the Rational Root Theorem

This theorem helps us find potential rational roots. For a polynomial P(x)=anxn+an1xn1++a0P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0, the potential rational roots are of the form:

factors of a0factors of an\frac{\text{factors of } a_0}{\text{factors of } a_n}

Here, an=3a_n = -3 (leading coefficient) and a0=21a_0 = -21 (constant term).

Factors of 21-21: ±1,±3,±7,±21\pm 1, \pm 3, \pm 7, \pm 21

Factors of 3-3: ±1,±3\pm 1, \pm 3

Potential rational roots: ±1,±13,±3,±7,±73,±21\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=1x = 1:

P(1)=3(1)3+5(1)2+19(1)21=3+5+1921=0P(1) = -3(1)^3 + 5(1)^2 + 19(1) - 21 = -3 + 5 + 19 - 21 = 0

So, x=1x = 1 is a root.

Step 3: Polynomial Division

Now, we divide P(x)P(x) by (x1)(x - 1) using synthetic or long division.

After dividing 3x3+5x2+19x21-3x^3 + 5x^2 + 19x - 21 by x1x - 1, the quotient is:

3x2+2x+21-3x^2 + 2x + 21

Step 4: Solve the quadratic equation

Now, solve 3x2+2x+21=0-3x^2 + 2x + 21 = 0 using the quadratic formula:

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

For 3x2+2x+21-3x^2 + 2x + 21, we have a=3a = -3, b=2b = 2, and c=21c = 21.

x=2±(2)24(3)(21)2(3)=2±4+2526=2±2566x = \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=2±166x = \frac{-2 \pm 16}{-6}

Thus, the solutions are:

x=2+166=146=73x = \frac{-2 + 16}{-6} = \frac{14}{-6} = -\frac{7}{3} x=2166=186=3x = \frac{-2 - 16}{-6} = \frac{-18}{-6} = 3

Step 5: List the solutions

The zeros of the polynomial are: x=1,x=73,x=3x = 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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Cubic Equations

Formulas

Rational Root Theorem: potential roots = ±(factors of constant term) / (factors of leading coefficient)
Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a

Theorems

Rational Root Theorem
Quadratic Formula

Suitable Grade Level

Grades 10-12