VIDEO: Factoring with binomial formulas

Factoring - you should know that

• You probably know the term "factor" from multiplication, because it is where two (or more) factors are multiplied together to get the product.
• A factor is therefore part of a multiplication problem, regardless of whether it comes from Counting or more complicated algebraic terms.
• If the task is "factorize", this means that the given term is broken down into individual factors. should be split up. In other words, you should make a multiplication out of it.
• If you are now to factor with binomial formulas, this means that you should create the binomial formulas in brackets from the given term. Incidentally, this corresponds to the inverse task of most Exercises with the binomial formulas, so to speak "formulas backwards".

Back to the binomial formulas - here's how

The prerequisite for factoring with binomial formulas is of course that you use these important formulas of the algebra master, in other words: be able to dissolve. Factoring then works according to the following scheme:

1. Use the two- or three-part expression given to determine which of the three formulas you are dealing with. You can recognize the first two binomial formulas by the sign of the mean term! The third binomial formula is only divided into two parts, so it can be easily recognized.
2. Determine the two substitutes a and b from the formula by finding numbers or letter combinations that, when squared, give the corresponding terms in the problem. Alternatively, you can also form the root of the first and last part of the term.
3. Then write the binomial formula in brackets.
4. Be sure to check the correctness of the solution. This last part is especially important for the first two binomial formulas, since the middle term (2ab) must be consistent (example below).

Binomial formulas backwards - examples for factoring

The rather dry approach should be explained using a few examples and a counterexample:

• You should convert the expression x² - 4xy + 4y² into a binomial formula. It is the second binomial formula (minus in the middle part). This has the form (a - b) ² and you will find a = x and b = 2y. Correspondingly, x² - 4xy + 4y² = (x - 2y) ². You still have to check the mean term 2ab = 2x_*2y = 4xy, so the result is correct.
• The expression 4y² + 4y + 64 initially looks as if it were the first binomial formula (2y + 8) ². However, checking the mean term shows that 2ab = 2y_*8 = 16y. So it is not a (!) Binomial formula. The expression cannot be factored (in this form).
• With the expression 4y⁴ - 25x⁸ it is about the third binomial formula (because it has two parts), which has the form (a + b) (a - b). You find a = 2y² and b = 5x⁴ and thus 4y⁴ - 25x⁸ = (2y² + 5x⁴) (2y² - 5x⁴). There is no testing here, as there is no central part.
• But be careful: The expression 40x³ - y² looks like the third binomial formula. However, the root cannot be drawn from 40x³. This term cannot be factored with binomial formulas either. Terms of the form x² + y² are also unsuitable, since the arithmetic symbol of the third binomial formula is incorrect.
• In some tasks, however, the formula "hides". With the expression 8x³ - 50x one would not initially assume a binomial formula. However, if you first factor out 2x (this is also factoring) and get 8x³ - 50x = 2x (4x² - 25), then the part of the brackets can then be converted into the third binomial formula. The result of this example is: 8x³ - 50x = 2x (2x + 5) (2x - 5). So if you come across an unsuitable candidate, the first thing to do is to check whether you can factor out one term first before converting the rest into one of the binomial formulas!