Special Factors

Rabu, 28 Juni 2023

In this lesson, you will learn to recognize and factor polynomials that are perfect square trinomials. They have two equal binomial factors.

The model below shows the product (x + 3)².

You can also use the FOIL method to find the product.

The square of (x + 3) is the sum of

·        the square of the first term of the binomial,

·        the square of the last term of the binomial, and

·        twice the product of the terms of the binomial.

These observations will help you recognize when a trinomial is a perfect square trinomial. They can be factored as shown.

Example

Determine whether each trinomial is a perfect square trinomial. If so, factor it.

 

1.     x² + 10x + 25

Alternative Solutions:

To determine whether x² + 10x + 25 is a perfect square trinomial, answer each question.

 

Therefore, x² + 10x + 25 is a perfect square trinomial.

x² + 10x + 25 = (x + 5)²

 

2.     4n² – 4n + 1 

Alternative Solutions:

 

 

Therefore, 4n² – 4n + 1 is a perfect square trinomial.

4n² – 4n + 1 = (2n – 1)²

 

3.     4p² – 12p + 36

Alternative Solutions:

 

Therefore, 4p² – 12p + 36 is not a perfect square trinomial.

 

Example

Geometry Link

 

4.     The area of a square is x² + 18x + 81. Find the perimeter.

Alternative Solutions:

Factor x² + 18x + 81 to find the measure of one side of the square.

x² + 18x + 81 = (x + 9)²

The measure of one side of the square is x + 9. A square has four sides of equal length. So, the perimeter is four times the length of a side.

The perimeter of the square is 4x + 36.

 

A polynomial like x² – 9 is called the difference of squares. Although this is not a trinomial, it can be factored into two binomials. The model shows how to factor x² – 9.

A difference of squares can be factored as shown.

Example

Determine whether each binomial is the difference of squares. If so, factor it.

 

5.     a² – 25

Alternative Solutions:

a² and 25 are both perfect squares, and a² – 25 is a difference.

 

6.     y² + 100

Alternative Solutions:

y² and 100 are both perfect squares. But y² + 100 is a sum, not a difference. Therefore y² + 100 is not a difference of squares. It is a prime polynomial.

 

7.     3n² – 48

Alternative Solutions:

First, look for a GCF. Then, determine whether the remaining factor is a difference of squares.

 

The following chart summarizes factoring methods.

 

Sumber

Labels: Mathematician

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