Unformatted text preview: A BETTER GENERAL FACTORING STRATEGY First, categorize the polynomial as a Binomial, Trinomial, or 4 Term Polynomial… Factoring Binomials Factoring Trinomials Order While one might be able to factor 6 + x2 + 5x intuitively without re‐ordering the trinomial, all of the formal methods introduced in this course require the trinomial be in descending or ascending order by If the First Coefficient is Negative degree; e.g.: If the coefficient of the first term is positive, skip Descending order (2‐1‐0): x2 + 5x + 6 forward; otherwise: Ascending order (2‐3‐4): x2 + 5xyz + 6y2z2 If the coefficients of the first and second terms are Note: If the degree of each term is equal, as in negative, factor out (–1); e.g., –x3 – 8 = (–1)(x3 + 8) 4x4 + 12x2y2 + 9y4, while the trinomial is not in If the coefficient of the first term is negative (and descending or ascending order, you can tell it is in the the coefficient of the second term is positive), either: correct order because the term with both x and y o Reorder the binomial; e.g., –9 + x2 = x2 – 9, or variables is in between the “x‐only” and “y‐only” term. o Factor out (–1); e.g., –x3 + 8 = (–1)(x3 – 8) If the First Coefficient is Negative Common Factors If the coefficient of the first term is negative, factor Be sure to factor out all common factors, including out (–1) constants, i.e., numbers, and variables Common Factors Available Factoring Methods Be sure to factor out all common factors, including 2
2 Sum of Squares: A + B = Prime/Not Factorable constants, i.e., numbers, and variables Difference of Squares: A2 – B2 = (A + B)(A – B) Available Factoring Methods Sum of Cubes: A3 + B3 = (A + B)(A2 – AB + B2) If you can easily factor the trinomial intuitively, you Difference of Cubes: A3 – B3 = (A – B)(A2 + AB + B2) should; otherwise: Order The only reason to re‐order a binomial is to make factoring more convenient; e.g., if the coefficient of the first term is negative (see next). Special Case A6 – B6: Difference of Squares OR Difference of Cubes
o Do Difference of Squares first… A6 – B6 = (A3 + B3)(A3 – B3) o Then Sum of Cubes and Difference of Cubes … (A3 + B3)(A3 – B3) = (A + B)(A2 – AB + B2) (A – B)(A2 + AB + B2) Trinomial Square: Check to see if the trinomial is a Trinomial Square, and if so, factor accordingly. a = 1: If the coefficient of the first term = 1, use the C Method to factor the trinomial. a ≠ 1: If the coefficient of the first term ≠ 1: o Use the AC Method to split the middle term (converting your trinomial into a 4 term polynomial), and o Use Factor by Grouping to factor the new 4 term polynomial. Factoring 4 Term Polynomials Order The only reason to re‐order a 4 Term Polynomial is to make factoring more convenient; e.g.: Given 14x3 + 15x2 + 28x + 30, you might simply find factoring (14x3 + 28x) + (15x2 + 30) easier than factoring (14x3 + 15x2) + (28x + 30), and If the middle operation in a 4 Term Polynomial is subtraction, such as in x3 + 7x – 6x2 – 42, you might find factoring (x3 – 6x2) + (7x – 42) easier than con‐
verting “subtraction” to “addition of the opposite” and factoring (x3 + 7x) + (–6x2 – 42) If the First Coefficient is Negative If the coefficient of the first term is negative, factor out (–1) Common Factors Be sure to factor out all common factors, including constants, i.e., numbers, and variables Available Factoring Methods Factor by Grouping Remember: If the “Reality Check” fails, you either: (i) Made a mistake when executing the method, or (ii) The failed “Reality Check” indicates the 4 Term Polynomial is Prime or Not Factorable. Always remember to (i) factor completely, and (ii) check by answer by multiplying all together to see if you get your original polynomial ...
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