4.9 Review and Summary
Additional Information
Access these online video resources for additional instruction and practice with factoring polynomials.
Key Equations
| Perfect square trinomial | [latex]\left(x+a\right)^2=\left(x+a\right)\left(x+a\right)=x^2+2ax+a^2[/latex] [latex]a^2+2ab+b^2=\left(a+b\right)^2[/latex] |
|---|---|
| Difference of squares | [latex]\left(a+b\right)\left(a-b\right)=a^2-b^2[/latex] [latex]a^2-b^2=\left(a+b\right)\left(a-b\right)[/latex] |
| Sum of cubes | [latex]a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)[/latex] |
| Difference of cubes | [latex]a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)[/latex] |
Key Terms
Binomial – a polynomial containing two terms
Coefficient – any real number [latex]a_i[/latex] in a polynomial in the form [latex]a_nx^n+...+a_2x^2+a_1x+a_0[/latex]
Difference of Squares – the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign
Distributive Property – the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, [latex]a⋅(b+c)=a⋅b+a⋅c[/latex]
Factor by Grouping – a method for factoring a trinomial in the form [latex]ax^2+bx+c[/latex] by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression
Greatest Common Factor – the largest polynomial that divides evenly into each polynomial
Leading Coefficient – the coefficient of the leading term
Leading Term – the term containing the highest degree
Perfect Square Trinomial – the trinomial that results when a binomial is squared
Trinomial – a polynomial containing three terms
Key Concepts
- The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See 4.2 Factoring the Greatest Common Factor of a Polynomial.
- Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See 4.3 Factoring a Trinomial with Leading Coefficient 1.
- Trinomials can be factored using a process called factoring by grouping. See 4.4 Factoring by Grouping.
- Perfect square trinomials and the difference of squares are special products and can be factored using equations. See 4.6 Factoring a Difference of Squares.
- The sum of cubes and the difference of cubes can be factored using equations. See 4.7 Factoring the Sum and Difference of Cubes.
- Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See 4.8 Factoring Expressions with Fractional or Negative Exponents.
Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites
