4.9 Review and Summary

Erin Kox

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	${(x + a)}^{2} = (x + a) (x + a) = x^{2} + 2 a x + a^{2}$ $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$
Difference of squares	$(a + b) (a - b) = a^{2} - b^{2}$ $a^{2} - b^{2} = (a + b) (a - b)$
Sum of cubes	$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$
Difference of cubes	$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$

Key Terms

Binomial – a polynomial containing two terms

Coefficient – any real number $a_{i}$ in a polynomial in the form $a_{n} x^{n} + . . . + a_{2} x^{2} + a_{1} x + a_{0}$

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, $a \cdot (b + c) = a \cdot b + a \cdot c$

Factor by Grouping – a method for factoring a trinomial in the form $a x^{2} + b x + c$ 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

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Additional Information

Key Equations

Key Concepts

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