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Квадраттық теңдеулерді жіктеу: Квадраттардың айырымы

Learn how to factor quadratics that have the "difference of squares" form. For example, write x²-16 as (x+4)(x-4).
Көпмүшелерді көбейткіштерге жіктеу үшін оларды екі немесе одан көп көпмүшелердің көбейтіндісі түріне келтіру керек. Бұл көпмүшелерді көбейтуге кері амал болып табылады.
In this article, we'll learn how to use the difference of squares pattern to factor certain polynomials. If you don't know the difference of squares pattern, please check out our video before proceeding.

Intro: Difference of squares pattern

Every polynomial that is a difference of squares can be factored by applying the following formula:
a2b2=(a+b)(ab)
Note that a and b in the pattern can be any algebraic expression. For example, for a=x and b=2, we get the following:
x222=(x+2)(x2)
The polynomial x24 is now expressed in factored form, (x+2)(x2). We can expand the right-hand side of this equation to justify the factorization:
(x+2)(x2)=x(x2)+2(x2)=x22x+2x4=x24
Now that we understand the pattern, let's use it to factor a few more polynomials.

Example 1: Factoring x216

Both x2 and 16 are perfect squares, since x2=(x)2 and 16=(4)2. In other words:
x216=(x)2(4)2
Since the two squares are being subtracted, we can see that this polynomial represents a difference of squares. We can use the difference of squares pattern to factor this expression:
a2b2=(a+b)(ab)
In our case, a=x and b=4. Therefore, our polynomial factors as follows:
(x)2(4)2=(x+4)(x4)
We can check our work by ensuring the product of these two factors is x216.

Тақырып бойынша біліміңді тексер

1) Factor x225.
Дұрыс жауапты таңдаңыз:

2) Factor x2100.

Ойлануға арналған сұрақ

3) Can we use the difference of squares pattern to factor x2+25?
Дұрыс жауапты таңдаңыз:

Example 2: Factoring 4x29

The leading coefficient does not have to equal to 1 in order to use the difference of squares pattern. In fact, the difference of squares pattern can be used here!
This is because 4x2 and 9 are perfect squares, since 4x2=(2x)2 and 9=(3)2. We can use this information to factor the polynomial using the difference of squares pattern:
4x29=(2x)2(3)2=(2x+3)(2x3)
A quick multiplication check verifies our answer.

Тақырып бойынша біліміңді тексер

4) Factor 25x24.
Дұрыс жауапты таңдаңыз:

5) Factor 64x281.

6) Factor 36x21.

Күрделі есептер

7*) Factor x49.

8*) Factor 4x249y2.