Factor The Sum And Difference Of Two Cubes – Presentation on the topic: “A3 4.1e Factoring the sum of two cubes and the difference of two cubes”— Presentation transcript:
3 rules of factoring! Always check if you can choose a common term. Count Terms!!! Two terms: a.) Difference of two squares: a2 – b2 = (a + b) (a – b) b.) Sum of two cubes: a3 + b3 = (a + b) (a2 – ab + b2) c . .) Differences of two cubes: a3 – b3 = (a – b) (a2 + ab + b2)
Factor The Sum And Difference Of Two Cubes
Lesson: Remember to have a list of perfect dice on your dice sheet. The formulas are: sum of two cubes: a 3 + b 3 = (a + b) (a 2 – ab + b 2) difference of two cubes: a 3 – b 3 = (a – b) (a 2 + ab + B 2) First decide what “a” and “b” are. Example: x3 + 8 Try: x6 + y6 a = x & b = 2 a = x2 & b = y2 (a + b) (a2 – ab + b2) (a + b) (a2 – ab + b2) (x ) + 2) (x2 – 2x + 4) (x2 + y2) (x4 – x2y2 + y4)
Solved: Sum And Difference Of Two Cubes D. Factor Each Of The Following: 1. Z^3 27 2. M^3+1 3. 8x^ [algebra]
The formulas are: sum of two cubes: a 3 + b 3 = (a + b) (a 2 – ab + b 2) difference of two cubes: a 3 – b 3 = (a – b) (a 2 + ab + B 2) First decide what “a” and “b” are. Example: x6 – 125 Try: x3 – y9 a = x2 & b = 5 a = 2x & b = y3 (a – b) (a 2 + ab + b 2) (a – b) (a 2 + ab + b 2) (x2) – 5) (x4 + 5×2 + 25) (2x – y3) (4×2 + 2×3 + y6)
Remember that the formulas are: Sum of two cubes: a 3 + b 3 = (a + b) (a 2 – ab + b 2) Difference of two cubes: a 3 – b 3 = (a – b) (a 2 + AB + B 2) Find out what “A” and “B” are. x9 – y15 a = 5×3 & b = y5 (5×3 – y5) (25×6 + 5x3y5 + y10) x y18 2 (8x y18) a = 2×4 & b = 3y6 2 (2×4 + 3y6) (4×8 – 6×12) Sum or Difference of 2 Odds Worksheet Cubes.
In order for this website to function, we record user data and share it with processors. To use this website, you must agree to our privacy policy, including our cookie policy. In Algebra class, the teacher would always discuss the topic of the sum of two cubes and the difference of two cubes side by side. The reason is that they are similar in structure. The key is to “memorize” or remember the patterns involved in the formulas.
Here are the formulas that summarize how to factor the sum and difference of two cubes. Study them carefully.
Solved: Factor Completely Each Sum And Difference Of Two Cubes. 10. M^3 64 11. 27+8p^3 12. 125+8q^ [algebra]
Rewrite the original problem as the sum of two cubes and then simplify. Since this is a “sum” example, the binomial factor and the trinomial factor will have positive and negative mean signs, respectively.
Use the rule for the difference of two cubes and simplify. Since this is a “difference” case, the binomial factor and trinomial factor will have negative and positive mean signs, respectively.
At first, this problem may seem “difficult”. However, if you stick with what we already know about the sum and the difference of two cubes, you should recognize that this problem is quite easy.
Sometimes it seems that a problem cannot be factored by the sum or difference of two cubes. If you notice something like this, try to remove common causes. For numbers, the greatest common factor is [latex]3[/latex], and for variables, the greatest common factor is “[latex]xi[/latex]”. Therefore, the general common factor would be their product, which is [latex] left (3 right) left ( right) = 3xi [/latex].
Ii. Factor Completely Each Sum And Difference Of Two Cubes.
When we factor, you will see that we have a simple problem about the difference of two cubes. Section 8.7 Review of Factorization Methods: Difference of Two Squares; Sum and difference of two dice.
Presentation on theme: “Section 8.7 Review of Factoring Methods: Differences of Two Squares; Sums and Differences of Two Cubes.”— Presentation transcript:
1 Section 8.7 Review of Factoring Methods: Differences of Two Squares; Sum and difference of two dice
Recall the special product rule for multiplying the sum and difference of the same two terms: (A + B) (A – B) = A2 – B2. The binomial A2 − B2 is called the difference of two squares because A2 is the square of A and B2 is the square of B. Inverse this rule gives us a method to factor the difference of two squares. Factor the difference of two squares: To factor the square of the first quantity minus the square of the last quantity, multiply the first plus the last by the first minus the last. f2 − l2 = (f + l) (f – l)
Solved:factor The Sum Or Difference Of Two Cubes. 27 8 X^3
To factor the difference of two squares, it is useful to know the first twenty integers of a perfect square. For example, the number 400 is a perfect square because 400 = 202.
5 Example 1 Factor: 49×2 – 16 Strategy The terms of this binomial have no common factor (except 1). The only option is to try to factor it as the difference of two squares. Why, if the binomial is the difference of two squares, it can be factored with a special product.
49×2 – 16 is the difference of two s squares because it can be written as (7x)2 – (4)2. We can relate it to the rule for factoring the difference of two squares to find the factorization. We can verify this result by multiplication.
The binomial x3 + y3 is called the sum of two cubes because x3 represents the cube of x, y3 represents the cube of y, and x3 + y3 represents the sum of the cubes. Similarly, x3 – y3 is called the difference of two cubes. Factoring the Sum and Difference of Two Cubes To factor the cube of the first quantity and the cube of the last quantity, multiply the first and last quantities by the first squared, minus the first, times the last, plus the last squared. F3 + L3 = (F + L)(F2 − FL + L2) To factor the cube of the first quantity minus the cube of the last quantity, multiply the first minus the last by the first squared, plus the first and last, plus The last square. F3 – L3 = (F – L) (F2 + Fl + L2)
In Exercises 57–64, Factor Using The Formula For The Sum Or Diffe…
The number 64 is called a perfect cube because 43 = 64. If you want to factor the sum or difference of two cubes, it is useful to know the first ten whole numbers of the perfect cube:
You need to remember the rules for factoring the sum and the difference of two cubes. Note that each has the form (binomial) (trinomial) and that there is a relationship between the letters that appear in these forms.
10 Example 7 Factor: a3 + 8 Strategy We will write the binomial in a form that shows that it is the sum of two cubes. Why we can then use the rule to factor the sum of two cubes.
Since a3 + 8 can be written as a3 + 23, it is the sum of two cubes that factor as follows: a2 – 2a + 4 is not a factor, it is a prime number. Therefore, a 3 + 8 = (a + 2) (a 2 – 2 a + 4). We can check by multiplication. Check: (a + 2)(a2 – 2a + 4) = a3 – 2a2 + 4a + 2a2 – 4a + 8 = a This is the original binomial.
Detailed Lesson Plan In Special Products
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