Math Problem Statement
find all zeros of P(x)= -3x^3+5x^2+19x-21
Solution
To find the zeros of the polynomial , we must solve for where .
Step 1: Use the Rational Root Theorem
This theorem helps us find potential rational roots. For a polynomial , the potential rational roots are of the form:
Here, (leading coefficient) and (constant term).
Factors of :
Factors of :
Potential rational roots:
Step 2: Test potential roots using synthetic division or direct substitution
Let’s check :
So, is a root.
Step 3: Polynomial Division
Now, we divide by using synthetic or long division.
After dividing by , the quotient is:
Step 4: Solve the quadratic equation
Now, solve using the quadratic formula:
For , we have , , and .
Thus, the solutions are:
Step 5: List the solutions
The zeros of the polynomial are:
Would you like a detailed explanation for any part of this process?
Here are 5 related questions:
- How can we use synthetic division to simplify polynomial division?
- What are the steps for applying the Rational Root Theorem?
- How can we solve a quadratic equation using factoring instead of the quadratic formula?
- What is the significance of complex roots in polynomials?
- 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
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