Advanced Math / Nonlinear Equations One Variable
Polynomial Equations in One Variable
Polynomial equations are often solved by moving all terms to one side, factoring, and applying the zero product property. Some SAT-style items ask for a particular solution, a nonzero solution, or the number of real solutions shown by factored form.
Least you need to know
- If a product of factors equals zero, at least one factor must equal zero.
- A common factor such as $x$ can create the solution $x=0$.
- Factored form makes solutions visible.
- When a problem asks for a positive, negative, least, greatest, or nonzero solution, give only the requested value.
Key notation
- $x(x-5)(x+2)=0$ — A polynomial equation with solutions $0$, $5$, and $-2$
- $x^3=16x$ — A cubic equation that can be solved by moving all terms to one side and factoring
Worked example
- To solve $x^3=9x$, move all terms to one side: $x^3-9x=0$.
- Factor: $x(x^2-9)=0$, so $x(x-3)(x+3)=0$.
- The solutions are $-3$, $0$, and $3$.
Common mistakes
- Do not divide by $x$ unless the problem has ruled out $x=0$; otherwise, you may lose a solution.
- The degree of a polynomial does not guarantee that every solution is real.
- When using factored form, solve each factor equal to zero.
How to recognize it
- The problem asks for a solution to a polynomial equation.
- The equation contains a cubic term and a common factor.
- The problem asks for the number of real solutions from a factored equation.
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