Advanced Math / Equivalent Expressions
Equivalent Polynomial Expressions
Polynomial expressions can often be simplified or rewritten by distributing, combining like terms, factoring out common factors, and recognizing special products such as differences of squares.
Least you need to know
- Combine only like terms: terms with the same variable part and exponent.
- Factor out the greatest common factor before looking for other factoring patterns.
- The difference of squares pattern is $a^2-b^2=(a-b)(a+b)$.
- Factored form can reveal zeros, while standard form can reveal coefficients and intercepts.
Key notation
- $x^3-9x$ — A polynomial that can be factored as $x(x-3)(x+3)$
- $(x+2)(x^2-2x+4)$ — A product that expands to a cubic polynomial
Worked example
- To factor $2x^3-18x$, first factor out $2x$.
- $2x^3-18x=2x(x^2-9)$.
- Then factor the difference of squares: $2x(x-3)(x+3)$.
Common mistakes
- Do not cancel or factor terms that are not common to every term.
- A cubic may factor by first removing a common factor.
- Expanding a product of three factors is easiest if two factors are multiplied first.
How to recognize it
- The problem asks which expression is equivalent.
- The expression has a common factor in every term.
- The expression includes $x^2-a^2$ or a related difference of squares.
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