Work through Steps 1 - 3 on your own
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. Discuss with a partner Step 4: Explain why Steps 2 and 3 give the same result 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
Difference of Two Squares
An example of the difference of two squares pattern is \( x^2 - 16 \).
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.
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:
Are the following polynomials a Difference of Two Squares?
Example 1: \( x^2 - 4 \)
Let's check each of the 3 conditions.
Example 2: \( x^2 - 8 \)
Let's check each of the 3 conditions.
Example 3: \( x^2 + 16 \)
Let's check each of the 3 conditions.
Example 4: \( 4x^2 - 100 \)
Let's check each of the "3" conditions.
Perfect Square Trinomial
- Are there two terms?
- Are the two terms being subtracted ?
- Are the terms both perfect squares?
Are the following polynomials a Difference of Two Squares?
Example 1: \( x^2 - 4 \)
Let's check each of the 3 conditions.
- Are there two terms? YES!
- Are the two terms being subtracted? YES!
- Are both terms perfect squares? YES!
Example 2: \( x^2 - 8 \)
Let's check each of the 3 conditions.
- Are there two terms? YES!
- Are the two terms being subtracted? YES!
- Are both terms perfect squares? NO! \(8\) is not a perfect square.
Example 3: \( x^2 + 16 \)
Let's check each of the 3 conditions.
- Are there two terms? YES!
- Are the two terms being subtracted? NO!
Example 4: \( 4x^2 - 100 \)
Let's check each of the "3" conditions.
- Are there two terms? YES!
- Are the two terms being subtracted? YES!
- Are both terms perfect squares? YES! \( 4x^2 \) is a perfect square!
Perfect Square Trinomial
An example of the perfect square trinomial pattern is \( x^2 -10x +25 \).
We are going to learn to recognize a Perfect Square Trinomial. Watch the video below to see how the pattern is formed.
There are 3 conditions we will need to check.
Are the following polynomials a perfect square trinomial?
Example 5: \( x^2 + 8x + 16 \)
Let's check each of the 3 conditions.
So, YES this polynomial is a perfect square trinomial.
Example 6: \( 4x^2 - 36x + 81 \)
Let's check each of the 3 conditions.
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.
So, NO this not a perfect square trinomial.
Example 8: \( x^2 - 20x - 100 \)
Let's check each of the 3 conditions.
- Is it a trinomial?
- Are the first and third terms perfect squares?
- Is the middle term twice the product of the square root of the first term and the third term?
Are the following polynomials a perfect square trinomial?
Example 5: \( x^2 + 8x + 16 \)
Let's check each of the 3 conditions.
- Is it a trinomial? YES!
- Are the first and third terms perfect squares? Yes! x2 and 16 are both perfect squares.
- Is the middle term twice the product of the square root of the first term and the third term? YES!
So, YES this polynomial is a perfect square trinomial.
Example 6: \( 4x^2 - 36x + 81 \)
Let's check each of the 3 conditions.
- Is it a trinomial? YES!
- Are the first and third terms perfect squares? Yes! \( 4x^2 \) and \(81\) are both perfect squares.
- Is the middle term twice the product of the square root of the first term and the third term? YES!
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.
- Is it a trinomial? YES!
- Are the first and third terms perfect squares? Yes! \( x2 \) and \(4\) are both perfect squares.
- Is the middle term twice the product of the square root of the first term and the third term? NO!
So, NO this not a perfect square trinomial.
Example 8: \( x^2 - 20x - 100 \)
Let's check each of the 3 conditions.
- Is it a trinomial? YES!
- Are the first and third terms perfect squares? No! \( x^2 \) is a perfect square but \(-100\) is not.
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
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