Solve polynomial equations

Solving polynomial equations involves finding all real and, when appropriate, nonreal complex solutions by factoring, applying the zero-product property and quadratic formula, using polynomial division or the Rational Root Theorem, and interpreting zeros as x-intercepts of the graph. The understanding includes repeated roots and their multiplicities, while distinguishing roots from factors and recognizing that a degree- $n$ polynomial has at most $n$ real roots and exactly $n$ complex roots when counted with multiplicity; general formulas for arbitrary higher-degree polynomials and advanced numerical methods are not included.

Detailed Explanation: Solve polynomial equations

To solve a polynomial equation, first write it in the form

P(x)=0.

Then try to factor $P(x)$ . Once the polynomial is written as a product, use the zero-product property: if

A(x)B(x)=0,

then $A(x)=0$ or $B(x)=0$ .

Worked example

Solve

x^4-4x^3+5x^2-4x+4=0.

1. Find a possible rational root

The possible rational roots are factors of the constant term divided by factors of the leading coefficient:

\pm1,\ \pm2,\ \pm4.

Test $x=2$ :

2^4-4(2^3)+5(2^2)-4(2)+4=16-32+20-8+4=0.

Since the result is $0$ , $x=2$ is a root, so $(x-2)$ is a factor.

2. Divide by the factor

Using synthetic division,

Therefore,

x^4-4x^3+5x^2-4x+4=(x-2)(x^3-2x^2+x-2).

3. Factor the quotient

Group the terms in the cubic:

\begin{aligned} x^3-2x^2+x-2 &=(x^3-2x^2)+(x-2)\\ &=x^2(x-2)+1(x-2)\\ &=(x-2)(x^2+1). \end{aligned}

Thus the original equation becomes

(x-2)^2(x^2+1)=0.

4. Solve each factor

Using the zero-product property:

(x-2)^2=0 \quad\Rightarrow\quad x=2,

and

x^2+1=0 \quad\Rightarrow\quad x^2=-1 \quad\Rightarrow\quad x=\pm i.

Therefore, the solutions are

\boxed{x=2,\quad x=i,\quad x=-i}.

The root $x=2$ has multiplicity $2$ because the factor $(x-2)$ appears twice. The polynomial has degree $4$ , so it has four complex solutions when counted with multiplicity:

2,\ 2,\ i,\ -i.

Only $x=2$ is real, so the graph has one real $x$ -intercept, at $(2,0)$ .

Learn by doing: Solve polynomial equations

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Polynomial Inequalities - Expanded Quadratic - Intervals

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