Polynomials & Factoring Cheat Sheet

A printable reference covering polynomial vocabulary, operations, special products, factoring patterns, grouping, and the zero product property for grades 9-10.

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Polynomials are algebraic expressions made from terms with variables and whole-number exponents. This cheat sheet helps students recognize polynomial structure, combine like terms, multiply expressions, and factor efficiently. These skills are essential for simplifying algebra, solving equations, and preparing for quadratic functions. The main ideas include writing polynomials in standard form, using exponent rules, applying special product patterns, and choosing a factoring strategy. Important formulas include $a^2-b^2=(a-b)(a+b)$ , $a^2+2ab+b^2=(a+b)^2$ , and $a^2-2ab+b^2=(a-b)^2$ . Factoring often means reversing multiplication, then using the zero product property to solve equations.

Key Facts

A polynomial in one variable can be written in standard form as $a_nx^{n}+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ , where $n$ is a whole number and $a_n\neq 0$ .
The degree of a polynomial is the greatest exponent of the variable, such as degree $4$ for $7x^4-3x^2+9$ .
Like terms have the same variable part and exponent, so $5x^2-2x^2=3x^2$ .
When multiplying powers with the same base, add exponents: $x^m\cdot x^n=x^{m+n}$ .
The distributive property is $a(b+c)=ab+ac$ , and factoring uses the reverse form $ab+ac=a(b+c)$ .
The difference of squares pattern is $a^2-b^2=(a-b)(a+b)$ .
Perfect square trinomials factor as $a^2+2ab+b^2=(a+b)^2$ and $a^2-2ab+b^2=(a-b)^2$ .
The zero product property says if $ab=0$ , then $a=0$ or $b=0$ .

Vocabulary

Polynomial: A polynomial is an expression made of terms added or subtracted, where variable exponents are whole numbers.
Monomial: A monomial is a polynomial with one term, such as $6x^3$ .
Binomial: A binomial is a polynomial with two terms, such as $x+5$ .
Trinomial: A trinomial is a polynomial with three terms, such as $x^2+7x+10$ .
Greatest Common Factor: The greatest common factor is the largest factor shared by all terms in an expression.
Factor: A factor is a number or expression that is multiplied by another factor to make a product.

Common Mistakes to Avoid

Adding exponents when adding like terms is wrong because exponents stay the same during addition, so $3x^2+4x^2=7x^2$ , not $7x^4$ .
Forgetting to factor out the greatest common factor first can make factoring harder and may leave the answer incomplete.
Using the difference of squares on a sum is wrong because $a^2+b^2$ does not factor as $(a+b)(a-b)$ .
Dropping negative signs during grouping changes the expression, so $x^3-2x^2+3x-6$ must be grouped carefully as $x^2(x-2)+3(x-2)$ .
Setting only one factor equal to zero misses solutions because if $(x-4)(x+2)=0$ , both $x-4=0$ and $x+2=0$ must be solved.

Practice Questions

1 Simplify and write in standard form: $(3x^2-5x+4)+(2x^2+7x-9)$ .
2 Multiply and simplify: $(x+6)(x-3)$ .
3 Factor completely and solve: $x^2-9x+20=0$ .
4 Explain how you would decide whether to factor $x^2+10x+25$ as a perfect square trinomial, and describe the pattern you are looking for.