Special Factoring Perfect Squares Perfect Cubes Perfect Square
Special Factoring Perfect Squares Perfect Cubes Perfect Square Trinomials
Perfect Square A perfect square is the product of a number, variable, or expression multiplied by itself. In other words, it’s what you get when you square an expression. Let’s look at some perfect squares.
Perfect Square Expression Work Perfect Square
Perfect Squares You should memorize the first 15 perfect squares. Let’s make a list of the first 15 perfect squares.
Perfect Squares
Square Root The square root of a number is number you square to get the product.
Perfect Squares �Any variable with an EVEN exponent is a perfect square.
Square Root To find square root of a variable with an even exponent, DIVIDE the exponent by 2.
Square Root Find the square root.
Objective 1 Factoring the Difference of Two Perfect Squares
Difference of Two Squares Recall, difference means subtraction. The difference of two squares means subtracting two perfect square expressions. 2 2 a –b
Difference of Two Squares This can be factored into binomial times binomial. One binomial must have a plus sign. The other binomial must have a minus sign. 2 2 a –b
Difference of Two Squares This can be factored into binomial times binomial. One binomial must have a plus sign. The other binomial must have a minus sign. 2 2 a – b =(a+b)(a – b)
Example Is this the difference of two perfect squares? YES
Example We can factor it into binomial times binomial. One binomial is addition, the other is subtraction.
Example What do you square to get x 2?
Example What do you square to get 36?
Example Is this the difference of two perfect squares? YES
Example We can factor it into binomial times binomial. One binomial is addition, the other is subtraction.
Example What do you square to get 4 x 2? or What is the square root of 4 x 2?
Example What do you square to get 81 y 2?
Example
Example Is this the difference of two perfect squares? YES
Example We can factor it into binomial times binomial. One binomial is addition, the other is subtraction.
Example What do you square to get 25 x 2?
Example 2 What do you square to get 1?
Example
Example Is this the difference of two perfect squares? YES
Example We can factor it into binomial times binomial. One binomial is addition, the other is subtraction.
Example What do you square to get x 2?
Example What do you square to get 36 y 4?
Example
Example Is this the difference of two perfect squares? YES
Example
Example What do you square to get x 2?
Example What do you square to get 4/25?
Example Is this the difference of two perfect squares? NO!
Example You can’t factor a sum of two squares. NO!
Example
Example Is this the difference of two perfect squares? NO But, we can still factor. GCF
Example Is this the difference of two perfect squares? YES
Example
Objective 2 Factoring the Sum of Two Perfect Cubes Factoring the Difference of Two Perfect Cubes
Perfect Cube A perfect cube is the product of a number, variable, or expression multiplied 3 times. In other words, it’s what you get when you cube an expression. Let’s look at some perfect cubes.
Perfect Cube Expression Work Perfect Cube
Perfect Cubes You should memorize the first 10 perfect cubes. Let’s make a list of the first 10 perfect cubes.
Perfect Cubes
Cube Root The cube root of a number is the number you cube to get the product.
Perfect Cubes �Any variable with an exponent that is DIVISIBLE BY 3 is a perfect cube.
Cube Root To find cube root of a variable, DIVIDE the exponent by 3.
Cube Root Find the cube root.
Sum of Two Cubes Recall, sum means addition. The sum of two cubes means adding two perfect cube expressions. 3 3 a +b
Sum of Two Cubes This can be factored into binomial times trinomial. The binomial must have a plus sign. The trinomial must have a minus sign before the middle term. 3 3 a +b
Sum of Two Cubes This can be factored into binomial times trinomial. The binomial must have a plus sign. The trinomial must have a minus sign before the middle term. 3 3 2 2 a + b =(a+b)(a – ab + b )
Example Is this the sum of two perfect cubes? YES
Example We can factor it into binomial times trinomial. The binomial is addition, the middle term of trinomial is subtraction.
Example We will fill in the binomial first.
Example What do you cube to get a 3?
Example What do you cube to get 64 y 3?
Example We will use the binomial to fill in the trinomial.
Steps for completing trinomial 1. Square the first term. 1 st Term in Trinomial 2. Square the last term. 3 rd Term in Trinomial 3. Multiply the two terms. Middle Term in Trinomial
Example Square the 1 st term.
Example Square the last term.
Example Multiply the two terms.
Example
Example Is this the sum of two perfect cubes? YES
Example We can factor it into binomial times trinomial. The binomial is addition, the middle term of trinomial is subtraction.
Example We will fill in the binomial first.
Example What do you cube to get 64 c 3?
Example What do you cube to get 27 d 3?
Example We will use the binomial to fill in the trinomial.
Example Square the first term.
Example Square the last term.
Example Multiply the two terms.
Example
Example Is this the sum of two perfect cubes? YES
Example We can factor it into binomial times trinomial. The binomial is addition, the middle term of trinomial is subtraction.
Example What do you cube to get 8 x 3?
Example What do you cube to get y 3 z 3?
Example We will use the binomial to fill in the trinomial.
Example Square the first term.
Example Square the last term.
Example Multiply the two terms.
Example
Example Is this the sum of two perfect cubes? YES
Example WAIT ! ! ! There is a GCF. We need to factor out the GCF first.
Example Now, see if we can factor the binomial. Is it the sum of two perfect cubes? Yes, we can factor it.
Example Bring down the 8 Don’t forget that it is part of our answer.
Example Now let’s factor the binomial. Binomial times trinomial.
Example We will fill in the binomial first.
Example What do you cube to get 8 c 3?
Example What do you cube to get d 3?
Example Use the binomial to fill in the trinomial.
Example Square the first term.
Example Square the last term.
Example Multiply the two terms.
Example
Difference of Two Cubes Recall, difference means subtraction. The difference of two cubes means subtracting two perfect cube expressions. 3 3 a –b
Difference of Two Cubes This can be factored into binomial times trinomial. The binomial must have a minus sign. The trinomial must have all plus signs. 3 3 a –b
Difference of Two Cubes This can be factored into binomial times trinomial. The binomial must have a minus sign. The trinomial must have all plus signs. 3 3 2 2 a – b =(a-b)(a + ab + b )
Example Is this the difference of two perfect cubes? YES
Example We can factor it into binomial times trinomial. The binomial has subtraction. The trinomial has ALL addition.
Example We will fill in the binomial first.
Example What do you cube to get 8 x 3?
Example What do you cube to get 27?
Example Use the binomial to fill in the trinomial.
Example Square the first term.
Example Square the last term.
Example Multiply the two terms.
Example
Example Is this the difference of two perfect cubes? YES
Example We can factor it into binomial times trinomial. The binomial has subtraction. The trinomial has ALL addition.
Example We will fill in the binomial first.
Example What do you cube to get a 3 b 3?
Example What do you cube to get 27?
Example Use the binomial to fill in the trinomial.
Example Square the first term.
Example Square the last term.
Example Multiply the two terms.
Example
Objective 3 Factoring a perfect square trinomial
Review – Chapter 5 FOIL
Review – Chapter 5 Binomial squared Trinomial
Review – Chapter 6 REVERSE Binomial squared This is called a perfect square trinomial. Trinomial
Factoring Perfect Square Trinomial The Steps Check a and c to make sure they are perfect squares. 2) Check the sign of the middle term to determine if you use addition or subtraction. 3) Take square root of a and c. 4) Check your answer. (Use FOIL) 1)
Example a b c Are a and c perfect squares? YES
Example Is the middle term positive or negative? positive
Example What do you square to get 4 x 2?
Example What do you square to get 9?
Example Let’s check our answer using FOIL
Example Check: CORRECT
Example
Example a b c Are a and c perfect squares? YES
Example Is the middle term positive or negative? negative
Example What do you square to get 4 x 2?
Example What do you square to get 25?
Example Let’s check our answer using FOIL
Example Check: CORRECT
Example
Example a b c Are a and c perfect squares? YES
Example Is the middle term positive or negative? positive
Example What do you square to get x 2?
Example What do you square to get 36?
Example Check: WRONG
Example This is not a perfect square trinomial. We must factor into binomial times binomial.
Example
Example
Things to Remember Think about what your answer should look like. binomial times trinomial binomial squared
Things to Remember Make sure you follow the correct formula or procedure for the type of expression you are factoring. Difference of Squares Sum of Cubes Difference of Cubes Perfect Square Trinomial
Things to Remember Watch your positive and negative signs. �ALWAYS check for GCF first before factoring using any other method.
Objective 4 Factoring Completely
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes Trinomial Product of Two Binomials ( )( ) If number in front multiply ac and replace middle term. Perfect Square Trinomial Polynomial Factor by Grouping
Example Look for GCF 3 a
Example trinomial
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes Trinomial Product of Two Binomials ( )( ) If number in front multiply ac and rewrite middle term. Perfect Square Trinomial Polynomial Factor by Grouping
GCF Trinomial Product of Two Binomials If number in front multiply ac and rewrite middle term. Perfect Square Trinomial
Example Bring down the GCF.
Example
Example a b c Step 1: Multiply a times c
Example b of – 24 to get 5 ? Which can. Factors you combine 1 and – 24 – 1 and 24 4 and – 6 – 4 and 6 2 and – 12 3 and – 8 – 2 and 12 – 3 and 8
Example – 3 and 8 Replace middle term
Example Factor by Grouping
Example Factor out GCF from each group
Example
Example
Example Look for GCF No GCF
Example polynomial
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes Trinomial Product of Two Binomials ( )( ) If number in front multiply ac and rewrite middle term. Perfect Square Trinomial Polynomial Factor by Grouping
Example Factor GCF from each binomial
Example
Example One of these binomials can be factored.
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes Trinomial Product of Two Binomials( )( ) If number in front multiply ac and rewrite middle term. Perfect Square Trinomial Polynomial Factor by Grouping
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes
Example
Example
Example
Example Look for GCF No GCF
Example polynomial
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes Trinomial Product of Two Binomials ( )( ) If number in front multiply ac and rewrite middle term. Perfect Square Trinomial Polynomial Factor by Grouping
Example Factor GCF from each binomial
Example
Example One of these binomials can be factored.
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes Trinomial Product of Two Binomials ( )( ) If number in front multiply ac and rewrite middle term. Perfect Square Trinomial Polynomial Factor by Grouping
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes
Example
Example
Perfect Squares & Perfect Cubes VARIABLES Even Exponent Perfect Squares Divisible by 3 Perfect Cubes If. Ifyou youcan candivideexponentby by 2, 3, ititisisaaperfectsquare. cube.
Example 9 Look for GCF No GCF
Example 9 binomial
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes Trinomial Product of Two Binomials ( )( ) If number in front multiply ac and rewrite middle term. Perfect Square Trinomial Polynomial Factor by Grouping
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes
Example One of these binomials can be factored. It is the difference of 2 squares.
Example
Example
Example Look for GCF 5 x
Example Trinomial
GCF Binomial Diff of two squares Sum of two cubes Diff of two cubes Trinomial Product of Two Binomials ( )( ) If number in front multiply ac and rewrite middle term. Perfect Square Trinomial Polynomial Factor by Grouping
GCF Trinomial Product of Two Binomials ( )( ) If number in front multiply ac and rewrite middle term. Perfect Square Trinomial
Example
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