Polynomials-Target+D-Bi+and+Trinomials-Guided+Learning

Target Poly D - Recognize a difference of two squares or perfect square trinomial.
Step 1: Pick any two consecutive numbers. Step 2: Square each, and find the difference of the larger square minus the smaller square. Step 3: Add the two original numbers. Step 4: Explain why Steps 2 and 3 give the same result
 * [[image:Discuss this.png width="208" height="116"]] || **Work through Steps 1 - 3 on your own **
 * Discuss with a partner **

Adapted from NCTM Illuminations ||

We are factoring polynomials. We learned in Target Poly C that we need to look for a GCF before we begin to factor using any other method. In this section we will learn to recognize two special types of polynomials which will allow us to factor these polynomials quickly and efficiently.

__**Difference of Two Squares**__ x 2 - 16 This is exactly what it sounds like... Subtraction of two terms that are both perfect squares. (Remember perfect squares are numbers that when you take their square root you get an integer!!) Let's look at a video to see where this pattern comes from.

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It is important that we learn to recognize this special polynomial before we begin to factor it. When we determine if a polynomial is a difference of two squares there are "3" conditions to check: 1 - Are there two terms? 2 - Are the two terms being subtracted ? 3 - Are the terms both perfect squares? All "3" conditions must be met in order for the polynomial to be a difference of two squares. If one condition is a "no" then the polynomial is not a difference of two squares.

Are the following polynomials a __//Difference of Two Squares//__? Example 1: x 2 - 4

Let's check each of the "3" conditions: 1 - Are there two terms? YES! 2 - Are the two terms being subtracted? YES! 3 - Are both terms perfect squares? YES!

So **YES** this polynomial is a difference of two squares!

Example 2: x 2 - 8

Let's check each of the "3" conditions: 1 - Are there two terms? YES! 2 - Are the two terms being subtracted? YES! 3 - Are both terms perfect squares? NO! 8 is not a perfect square!

So **NO** this polynomial is not a difference of two squares!

Example 3: x 2 + 16

Let's check each of the "3" conditions" 1 - Are there two terms? YES! 2 - Are the two terms being subtracted? NO!

So **NO** this polynomial is not a difference of two squares! Since this is not a difference then this polynomial does not meet the conditions, the third condition does not even need to be addressed.

Example 4: 4x 2 - 100

Let's check each of the "3" conditions: 1 - Are there two terms? YES! 2 - Are the two terms being subtracted? YES! 3 - Are both terms perfect squares? YES! 4x 2 is a perfect square!

So **YES** this polynomial is a difference of two squares!

__**Perfect Square Trinomial**__ x 2 - 10x + 25 We are going to learn to recogniz e a Perfect Square Trinomial. Watch the video below to see how the pattern is formed.

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 * There are "3" conditions we will need to check. **

1 - Is it a trinomial? 2 - Are the first and third terms perfect squares? 3 - Is the middle term twice the product of the square root of the first term and the third term? All "3" terms must be met in order for the polynomial to be a perfect square trinomial. If just one condition is not met, then the polynomial is not a perfect square polynomial.

Are the following polynomials a //perfect square trinomial//?


 * Example 5: x 2 + 8x + 16**

Let's check each of the "3" conditions: 1 - Is it a trinomial? YES! 2 - Are the first and third terms perfect squares? Yes! x 2 and 16 are both perfect squares. 3 - Is the middle term twice the product of the square root of the first term and the third term? YES! Let's see how: the square root of x 2 is x and the square root of 16 is 4. And 4 times x is 4x and 2 times 4x is 8x - which is the same as the middle term!

So **YES** this polynomial is a perfect square trinomial!


 * Example 6: 4x 2 - 36x + 81**

Let's check each of the "3" conditions: 1 - Is it a trinomial? YES! 2 - Are the first and third terms perfect squares? Yes! 4x 2 and 81 are both perfect squares. 3 - Is the middle term twice the product of the square root of the first term and the third term? YES! Let's see how: the square root of 4x 2 is 2x and the square root of 81 is 9. And 2x times 9 is 18x and 2 times 18x is 36x - which is the same as the middle term! Now what about the middle term being negative - that is ok!

So **YES** this is perfect square trinomial!


 * Example 7: x 2 + 2x + 4**

Let's check each of the "3" conditions: 1 - Is it a trinomial? YES! 2 - Are the first and third terms perfect squares? Yes! x 2 and 4 are both perfect squares. 3 - Is the middle term twice the product of the square root of the first term and the third term? NO! Let's see why: the square root of x 2 is x and the square root of 4 is 2. And 2 times x is 2x and twice 2x is 4x - which is the not the same as the middle term.

So **NO** this not a perfect square trinomial.


 * Example 8: x 2 - 20x - 100 **

Let's check each of the "3" conditions: 1 - Is it a trinomial? YES! 2 - Are the first and third terms perfect squares? No! x 2 is a perfect square but -100 is not.

So **NO** this not a perfect square trinomial.


 * Quick Check - Polynomial - Target D - Quick Check: ****

Are the following a difference of two squares?

1) x 2 - 25

2) 16x 2 + 64

Are the following a perfect square trinomial?

3) 4x 2 - 12x + 36

4) x 2 - 10x + 25

Quick Check Solutions Polynomials - Target D - Bi and Trinomials