Problem
The keyword “zeros of polynomial calculator” points to the task of finding the zeros (roots) of a polynomial. Consider the cubic polynomial \(P(x) = x^3 - 4x^2 + x + 6\). Determine all zeros of \(P(x)\) and interpret them on the graph.
Solution
1) What “zeros of a polynomial” means
A number \(r\) is a zero of \(P(x)\) if it satisfies \(P(r)=0\). Solving \(P(x)=0\) finds all zeros (real and complex). Each real zero corresponds to an x-intercept of the graph \(y=P(x)\).
2) Use the Rational Root Theorem to list candidates
For \(P(x)=x^3-4x^2+x+6\), the leading coefficient is \(1\) and the constant term is \(6\). The Rational Root Theorem says any rational zero must be among the factors of \(6\): \(\pm 1, \pm 2, \pm 3, \pm 6\).
3) Test candidates and identify a first zero
Evaluate \(P(x)\) at a simple candidate:
Therefore, \(x=-1\) is a zero, so \((x+1)\) is a factor (Factor Theorem).
4) Divide out the factor using synthetic division
Divide \(P(x)\) by \((x+1)\) to reduce the cubic to a quadratic. Synthetic division with \(r=-1\) (coefficients \(1, -4, 1, 6\)) gives:
| Step | Work | Result |
|---|---|---|
| Coefficients | 1, -4, 1, 6 | — |
| Bring down | 1 | 1 |
| Multiply by \(-1\) | 1 · (-1) = -1 | add to -4 → -5 |
| Multiply by \(-1\) | -5 · (-1) = 5 | add to 1 → 6 |
| Multiply by \(-1\) | 6 · (-1) = -6 | add to 6 → 0 (remainder) |
The quotient polynomial is \(x^2 - 5x + 6\), and the remainder is \(0\). Hence:
5) Factor the quadratic to find the remaining zeros
Factor \(x^2 - 5x + 6\) by finding two numbers whose product is \(6\) and sum is \(-5\): \(-2\) and \(-3\).
Therefore, the full factorization is:
Zeros (roots): \(x=-1\), \(x=2\), \(x=3\).
Each zero has multiplicity \(1\) (each factor appears once), so the graph crosses the x-axis at each intercept.
6) Quick verification
Substituting each zero into \(P(x)\) confirms \(P(-1)=0\), \(P(2)=0\), and \(P(3)=0\). Since the polynomial is degree \(3\), exactly three zeros counting multiplicity are expected; all are real here.
7) General note: when factoring does not finish the job
A zeros of polynomial calculator often combines symbolic steps (factoring, synthetic division) with general formulas. For example, if a quadratic factor \(ax^2+bx+c\) remains unfactored over the integers, its zeros are:
For real-coefficient polynomials, any non-real complex zeros occur in conjugate pairs.
Visualization
Final result
\(P(x)=x^3-4x^2+x+6\) factors as \((x+1)(x-2)(x-3)\), so the zeros are \(x=-1\), \(x=2\), and \(x=3\), which are exactly the x-intercepts of the graph.