Factoring using contracted multiplication

🏆Practice decomposition of factors according to the abbreviated multiplication formulas

The formulas for shortened multiplication will help us convert expressions with terms that have among them signs of addition or subtraction into expressions whose terms are multiplied.
The formulas for contracted multiplication are:
a2−b2=(a−b)×(a+b)a^2-b^2=(a-b)\times (a+b)

a2+2ab+b2=(a+b)2a^2+2ab+b^2=(a+b)^2

a2−2ab+b2=(a−b)2a^2-2ab+b^2=(a-b)^2

To use the first formula: a2−b2=(a−b)×(a+b) a^2-b^2=(a-b)\times(a+b)

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Test yourself on decomposition of factors according to the abbreviated multiplication formulas!

einstein

Find the value of the parameter x.

\( x^2+x=0 \)

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We will ask ourselves:

  1. Is there a positive and a negative expression? 
  2. Can we find the root of each of the expressions separately? 


If we have answered positively to both questions, all that remains for us to do is simply take the root of the two terms and write them according to the formula.

Observe:
We will place the roots in parentheses.
In case there are two positive terms or two negative terms, it will not be possible to use this formula.

To use the other formulas :a2+2ab+b2=(a+b)2a^2+2ab+b^2=(a+b)^2

we must verify that three conditions are met. We will ask ourselves:

  1. Do the two terms from which we will take the roots have the same sign? That is, are both positive or both negative?
  2. Can we take the root of the two terms separately? a a and B B ? 
  3. If we multiply the product of the roots by 2 2 will we obtain the middle term (in positive or in negative)? 

If we have answered positively to all the questions,
all that remains for us to do is simply place the obtained roots in the corresponding formula.
Observe that, if the third term was negative in the original exercise we will place it in the formula with the subtraction sign. When can these formulas not be used?
When in the original exercise the signs of the terms from which we want to take roots are different, that is, one term positive and the other negative, we cannot use these formulas.


Example of the use of the first abbreviated multiplication formula:

x2−64=x^2-64=
We will ask ourselves:

  • Is there a positive expression and a negative one? The answer is yes. 
  • Can we find the root of each of the expressions separately? The answer is yes.

Magnificent. All that remains for us to do is simply take the root of the two terms and write them according to the formula.
We will obtain:
(x−8)(x+8)(x-8)(x+8)


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Example of the use of these abbreviated multiplication formulas:

x2−18+81=x^2-18+81=
We will ask ourselves:

  • Do the two terms from which we will extract the roots have the same sign? That is, are both positive or both negative? The answer is yes. Both are positive.
  • Can we extract the root of the two terms separately? aa and bb Is the answer positive?

root aa will be xx
root bb will be 99

If we multiply the product of the roots by 22, will we obtain the middle term (in positive or in negative)? The answer is yes.
9×x×2=18x9\times x\times 2=18x

Magnificent. Now, all that remains for us to do is simply place the roots obtained in the corresponding formula.
Note that the middle term has the minus sign in the original exercise. Therefore, we will put it in the formula with a minus sign and obtain:
(x−9)2(x-9)^2


If you are interested in this article, you might also be interested in the following articles:

  • Factorization
  • The uses of factorization
  • Factorization through the extraction of the common factor outside the parentheses
  • Factorization of trinomials
  • Factorization of algebraic fractions
  • Addition and subtraction of algebraic fractions
  • Simplification of algebraic fractions
  • Multiplication and division of algebraic fractions
  • Solving equations through factorization

In the Tutorela blog, you will find a variety of articles about mathematics.


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