Factoring Polynomials

Sum of Cubes

A polynomial in the form a³ + b³ is called a SUM OF CUBES.

Difference of Cubes

A polynomial in the form a³ - b³ is called the DIFFERENCE OF CUBES.

Factoring Trinomials Formatted 'ax² + bx + c'

Copy and paste the following link into your browser to learn more about factoring trinomials of the form 'ax² + bx + c' : http://www.bing.com/videos/search?q=Trinomials+of+the+Form+%27ax²+%2b+bx+%2b+c%27&&view=detail&mid=A2E867C8E1F191B65905A2E867C8E1F191B65905&FORM=VRDGAR

Sequencing Difference of Squares and Diiference of Cubes

Factor a difference of squares BEFORE a difference of cubes. For example: x⁶ - y⁶ = (x³)² - (y³)² [Difference of Squares] = (x³ + y³)(x³ - y³) [Difference of Cubes]

Factoring Sequence for GCF and Difference of Squares

Many polynomials require more than one method of factoring to be completely factored into a product of polynomials. Because of this, a sequence of factoring methods must be used. ✔︎ First, try to factor by using the GCF. ✔︎ Second, try to factor by using the difference of squares.

Greatest Common Factor [GCF]

The first method of factoring is called factoring out the GCF (greatest common factor). For example, Factor 5 x + 5 y. Since each term in this polynomial involves a factor of 5, then 5 is a common factor of the polynomial.

Difference of Squares

The product of conjugates produces a pattern called a DIFFERENCE OF SQUARES. Factoring a difference of squares also requires its own set of steps. You can recognize a difference of squares because it's always a binomial, where both terms are perfect squares and a subtraction sign appears between them. It always appears as a² - b², or (something)² - (something else)².

Square Trinomial

Therefore, if a trinomial is of the form ( x) 2 + 2( x)( y) + ( y) 2, it can be factored into the square of a binomial. ➜ For example: Is 4x² - 20 x + 25 a square trinomial? If so, factor it into the square of some binomial. 4x² = (2x)² and 25 = (-5)² and -20x = 2(2x)(-5) So it is a square trinomial, which factors as follows. 4x² - 20x + 25 = (2x - 5)²

Factoring Polynomials

To FACTOR a polynomial means to rewrite the polynomial as a product of simpler polynomials or of polynomials and monomials. Because polynomials may take many different forms, many different techniques are available for factoring them.

Factoring Polynomials by Regrouping

To attempt to factor a polynomial of four or more terms with no common factor, first rewrite it in groups. Each group may possibly be separately factored, and the resulting expression may possibly lend itself to further factorization if a greatest common factor or special form is created. Copy and paste the following link into your browser to learn more about factoring polynomials by regrouping: http://www.bing.com/videos/search?q=%2bFactoring+Polynomials+by+Regrouping&&view=detail&mid=DEA4681F594F972722DADEA4681F594F972722DA&FORM=VRDGAR

Factoring Trinomials Formatted 'x² + bx + c'

To factor polynomials of the form x² + bx + c, begin with two pairs of parentheses with x at the left of each. (x )(x ) Next, find two integers whose product is c and whose sum is b and place them at the right of the parentheses. For example, factor x² + 8x + 12: x² + 8x + 12 = (x )(x ) The number 12 can be factored via several combinations:: (1)(12) or (-1)(-12) or (2)(6) or (-2)(-6) or (3)(4) or with (-3)(-4) HOWEVER, the ONLY combination whose sum is also equal to 8 is (2)(6), therefore, x² + 8x + 12 = (x + 2)(x + 6)

Factoring Polynomials and Solving Higher Degree Equations Using the Zero Product Rule

With respect to division polynomials behave a lot like natural numbers. This means it is not always possible to divide two polynomials and get a polynomial as a result. The result may sometimes be a polynomial but in general we will get a rational expression. This method uses the ZERO PRODUCT RULE, which specifies, if ( a)( b) = 0, then: ➜ Either ( a) = 0, ( b) = 0, or both. For example, solve x( x + 3) = 0. Apply the zero product rule. x = 0, or x + 3 = 0, then x = -3 Copy and paste the following link into your browser to learn more about factoring polynomials and solving higher degree equations, using the zero product rule: https://youtu.be/SDe-1lGeS0U

Summary of Factoring Techniques

✔︎ FOR ALL POLYNOMIALS, first factor out the greatest common factor (GCF). ✔︎ FOR A BINOMIAL, check to see if it is any of the following: ☛ difference of squares: x²- y² = ( x + y) ( x - y) ☛ difference of cubes: x³ - y³ = ( x - y) ( x² + xy + y²) ☛ sum of cubes: x³ + y³ = ( x + y)( x² - xy + y²) ✔︎ FOR A TRINOMIAL, check to see whether it is either of the following forms: ☛ x² + bx + c: If so, find two integers whose product is c and whose sum is b. For example, x² + 8 x + 12 = ( x + 2)( x + 6) since (2)(6) = 12 and 2 + 6 = 8 ☛ ax 2 + bx + c: If so, find two binomials so that the product of first terms = ax² the product of last terms = c the sum of outer and inner products = bx ✔︎ FOR POLYNOMIALS WITH FOUR OR MORE TERM, first regroup, factor each group, and then find a pattern.