什么是因子三项式公式？

三项式是具有三个项的多项式。三项式的示例是 x+y+z、x ² +2x+2、x+y-1 等。三项式可以有两种类型。它们是完美平方三项式和非完美平方三项式。分解多项式只不过是将表达式多项式写为两个或多个表达式的乘积。根据给定的三项式类型，遵循一组不同的步骤。

完美平方三项式的因式分解

有两个公式可以分解完美的平方三项式。这些在下面提到 -

a² + 2ab + b² = (a + b)²

a² – 2ab + b² = (a – b)²

应该记住，给定的三项式只有在其形式为² +2ab+b ²或² -2ab+b ²时才是完全平方三项式。

因式分解非完美平方三项式

当且仅当它的形式为 ax ² +bx+c 而不是完全平方三项式时，才称该三项式为非完全平方三项式。下面提到了分解这种三项式的步骤 -

1. Determine a, b, c in a trinomial and find ac value.
2. Find two numbers whose product is ac and sum is equal to b.
3. Split the middle term in a trinomial into sum of two terms using the two numbers found in the step-2.
4. Factor by grouping.

示例问题

问题 1：分解三项式 x ² + 4x + 4。

解决方案：

Given trinomial,

x²+4x+4

This can be written as-

x²+2(2)(x)+2²

It is in the form of a²+2ab+b² where a=x and b=2

So it is a perfect square trinomial and one of the formula of it can be applied in factoring.

a²+2ab+b²=(a+b)²

So, x²+4x+4=(x+2)²

(x+2)²=>(x+2)(x+2)

问题 2：使用因式分解三项式公式对给定多项式 x ² – 2x + 1 进行因式分解。

解决方案：

Given trinomial,

x²-2x+1

This can be written as-

x²-2(1)(x)+1²

It is in the form of a²+2ab+b² where a=x and b=1

So it is a perfect square trinomial and one of the formula of it can be applied in factoring.

a²-2ab+b²=(a-b)²

So, x²-2x+1=(x-1)²

(x-1)²=>(x-1)(x-1)

问题 3：分解三项式 x ² – 2x – 3。

解决方案：

Given trinomial,

x²-2x-3

This cannot be written into a²+2ab+b² or a²-2ab+b². So it is not a perfect square trinomial.

So need to follow the steps to factorize a non perfect square trinomial.

Step 1: Compare given trinomial with ax²+bx+c

Where a=1,b=-2 and c=-3

ac=1×-3=-3

Step 2: Pick two numbers such that product of two numbers is equal to ac and sum of those two number is equal to b.

Let it be 1,-3

Step 3: Split the middle term into sum of two terms using above two numbers.

x²+1x-3x-3=>x(x+1)-3(x+1)

                  =(x+1)(x-3)

So, x²-2x-3=(x+1)(x-3)

问题 4：使用因式分解三项式公式对给定多项式 3x ² – 7x – 6 进行因式分解。

解决方案：

Given trinomial,

3x²-7x-6

This cannot be written into a²+2ab+b² or a²-2ab+b². So it is not a perfect square trinomial.

So need to follow the steps to factorize a non perfect square trinomial.

Step 1: Compare given trinomial with ax²+bx+c

Where a=3,b=-7 and c=-6

ac=3×-6=-18

Step 2: Pick two numbers such that product of two numbers is equal to ac and sum of those two number is equal to b.

Let it be -9,2

Step 3: Split the middle term into sum of two terms using above two numbers.

3x²-7x-6=>3x²-9x+2x-6

                =3x(x-3)+2(x-3)

                =(3x+2)(x-3)

So, 3x²-7x-6=(3x+2)(x-3)

问题 5：分解给定的三项式 2x ² – 9x + 10。

解决方案：

Given trinomial,

2x²-9x+10

This cannot be written into a²+2ab+b² or a²-2ab+b². So it is not a perfect square trinomial.

So need to follow the steps to factorize a non perfect square trinomial.

Step 1: Compare given trinomial with ax²+bx+c

Where a=2,b=-9 and c=10

ac=2×10=20

Step 2: Pick two numbers such that product of two numbers is equal to ac and sum of those two number is equal to b.

Let it be -4,-5

Such that (-4)×(-5)=20 and -4+(-5)=-9

Step 3: Split the middle term into sum of two terms using above two numbers.

2x²-9x+10=>2x²-4x+(-5x)+10

                   = 2x²-4x-5x+10

                   = 2x(x-2)-5(x-2)

                   = (2x-5)(x-2)

So, 2x²-9x+10 = (2x-5)(x-2)