Polynomials Lesson 5Factoring Special Polynomials

Todays Objectives

Students will be able to demonstrate an understanding of the factoring of polynomial expressions, including: Factor a polynomial that is a difference of squares,

and explain why it is a special case of trinomial factoring where b = 0

Identify and explain errors in a solution for a polynomial expression

Factoring Special Polynomials

Today we will look at factoring two types of special polynomials Perfect square trinomials Difference of squares

Factoring Special Polynomials

Consider a square with side length a + b:a b

a

b

(𝑎)(𝑎)=𝑎2 (𝑎)(𝑏)=𝑎𝑏

(𝑏)(𝑎)=𝑎𝑏 (𝑏)(𝑏)=𝑏2

It’s area is: (a + b)2 = (a + b)(a + b)=a(a + b) + b(a + b)=a2 + ab + ab + b2

=a2 + 2ab + b2

Perfect Square Trinomials

We say that a2 + 2ab + b2 is a perfect square trinomial:

a2 is the square of the first term in the binomial

2ab is twice the product of the first and second terms in the binomial

b2 is the square of the second term in the binomial

When we use algebra tiles to represent a perfect square trinomial, the tiles will form a square shape.

Perfect Square Trinomials

2 forms of perfect square trinomials:

(a – b)2 = a2 – 2ab + b2

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

We can use these patterns to factor perfect square trinomials.

Example Factor the trinomial using your algebra tiles: 4x2 + 12x + 9

Solution:

Arrange the algebra tiles to form a square. The side lengths will be equal to 2x + 3. So we can see that this trinomial is a perfect square trinomial with factors (2x + 3)2.

Check:

(2x + 3)(2x + 3) = 2x(2x + 3) + 3(2x + 3)=4x2 + 6x + 6x + 9 = 4x2 + 12x + 9

The result is the same as our original trinomial, so the factors are correct

Example

Factor the trinomial 4 – 20x + 25x2

Solution:

The first term is a perfect square (4 = 2 x 2)

The third term is a perfect square (5x)(5x) = 25x2

The second term is twice the product of 5x and 2 (10x)(2) = 20x

Since the 2nd term is negative, the operations in the binomial factors must be subtraction.

So, the trinomial is a perfect square with factors: (2 – 5x)(2 – 5x) or (2-5x)2.

Example

Factor the trinomial 16 – 56x + 49x2

Solution:

(4 – 7x)2

Difference of Squares

Another example of a special polynomial is a difference of squares. A difference of squares is a binomial of the form a2 – b2. We can think of this as a trinomial with a middle term of zero. For example, we could write the perfect square (x2 – 25) as the trinomial (x2 – 0x – 25).

This is a perfect square because x2 = (x)(x), and 25 = (5)(5). Any subtraction expression is known as a difference. Therefor, this is a difference of squares.

Factoring a Difference of Squares

To factor this “trinomial”, we should find two integers whose product is -25, and whose sum is 0. These two integers are 5 and -5.

So, x2 – 25 = (x + 5)(x – 5).

This pattern is true for any difference of squares.

Example

Factor the difference of squares 25 – 36x2

Solution:

Write each term as a perfect square.

25 – 36x2 = (5)2 – (6x)2 = (5 + 6x)(5 – 6x)

Example

Factor 5x4 – 80y4

Solution:

As written in this example, each term is not a perfect square, but we can remove a common factor of 5.

= 5(x4 – 16y4)

= 5[(x2)2 – (4y2)2]

= 5(x2 – 4y2)(x2 + 4y2) The first binomial is also a difference of squares

= 5(x + 2y)(x – 2y)(x2 + 4y2)

Example

Factor the following:

81m2 – 49

Solution: (9m+7)(9m-7)

162v4 – 2w4

Solution: 2(81v4-w4) = (2)(9v2+w2)(9v2-w2)

=(2)(9v2+w2)(3v+w)(3v-w)