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FOM 11 Q 2 – FACTORING REVIEW 2 1

© R. Ashby 2019. Duplication by permission only.

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

1) FACTORS = numbers or variables multiplied together to create a term.

1) GREATEST COMMON FACTOR = the largest factor that is part of two or more different terms.

2) COMMON FACTOR = removing the greatest common factor from each term in a polynomial.

COMMON FACTORING I) COMMON FACTORING IS THE PROCESS OF REMOVING THE GREATEST COMMON FACTOR FROM EACH TERM IN A

POLYNOMIAL. When factoring a polynomial ALWAYS COMMON FACTOR FIRST!!! A) SAMPLE PROBLEMS 1: Study these examples carefully. Be sure you understand and memorize the process

used to complete them. 1) Factor 3x + 3y 2) Factor −3x 2−6x

Both terms have in common ∴ Both terms have in common ∴

3) Factor 10x 4y 2−25x 2y 3 4) Factor −6x 4y−15x 3y 2−3x 2y

Both terms have in common ∴ These terms have in common ∴

B) REQUIRED PRACTICE 1: Factor these expressions. SHOW THE PROCESS!! {Answers are on page 4 of these notes.} 1) 4x −12y 2) 5xy−10x 2y 3) 7x

3y 4 − 49x 2y 3 4) 24x 2y 3− 42xy 2

5) 32a3w2−64a2w 6) 10a3t 2−25at 2 7) 91gh2− 49g 2h 8) 12xy 2 + 27x 2y

9) 49x 2y 4 + 63x 3y 3 10) 3ab−6ac−9bc 11) 2xy− 4xw−6xw2 12) −44a−77b−33c

13) 15ax −30ay− 45aw 14) 5ax 3−10a2x −15a3x 15) −2a3x 2− 4a2x 3−8a2x 2

16) 6x2y +15xy 2−27xy 17) 36x 4y 2− 45x 2y 3−18xy 4 18) −35m2n3−21m2n2−56m2n

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) QUADRATIC = an expression, equation or function where the largest exponent on its variable(s) is a 2.

2) PERFECT SQUARE = the result of squaring a number or a variable. e.g. 32 = 9, 9 is a perfect square.

3) DIFFERENCE OF PERFECT SQUARES = a perfect square subtracted from a different perfect square. e.g. x

2−y 2( ); x 2− 4( ); 9x 2−25( ); 4x 2−36y 2( ); x 2−1( )

4) SUM OF PERFECT SQUARES = a perfect square added to a different perfect square. e.g. x

2 +y 2( ); x 2 + 4( ); 9x 2 + 25( ); 4x 2 + 36y 2( ); x 2 +1( )



RECOGNIZING QUADRATICS I) A QUADRATIC IS AN EXPRESSION, EQUATION OF FUNCTION WHERE THE LARGEST EXPONENT ON ITS

VARIABLE(S) IS A 2. The table below gives several examples of quadratics and the forms they can take. Be sure you understand and memorize their key characteristics as it is essential to be able to recognize them in order to solve them and use them appropriately.

QUADRATIC EXPRESSIONS QUADRATIC EQUATIONS QUADRATIC FUNCTIONS

x 2

x 2− 4x

3x 2 + 7x−4.9x 2 + 9−5x 2 +153.2x −9−4 x + 8( )2

−1

x 2 = 0x 2− 4x = 03x 2 + 7x = 0−4.9x 2 + 9 = 0−5x 2 +153.2x = 9−4 x + 8( )2

= 1

y = x 2

y = x 2− 4x

y = 3x 2 + 7x

y = −4.9x 2 + 9y = −5x 2 +153.2x −9

y = −4 x + 8( )2−1

A) REQUIRED PRACTICE 2: Page 360: Question 2. {Ans. Page 565}

FACTORING QUADRATIC BINOMIALS I) WHEN FACTORING POLYNOMIALS, ALWAYS COMMON FACTOR FIRST IF POSSIBLE!!

II) A DIFFERENCE OF PERFECT SQUARES IS A BINOMIAL CONSISTING OF A PERFECT SQUARE SUBTRACTED FROM A DIFFERENT PERFECT SQUARE.

e.g. x2− 4( ); x 2−1( ); 9x 2−25( ); x 2−y 2( ); 4x 2−36y 2( )

A) USE THESE STEPS TO FACTOR A DIFFERENCE OF PERFECT SQUARES1: COMMON FACTOR if possible. 2: Write two sets of brackets with the square root of the first term at the beginning of each set. 3: Write the square root of the second term at the end of each set of brackets. 4: Write a different sign, + or − , in each set of brackets.

B) SAMPLE PROBLEMS 3: Study these examples carefully. Be sure you understand and memorize the process used to complete them. 1) Factor x

2− 4 2) Factor x2−y 2 3) Factor 8x

2−18 4) Factor 5x 3−80x

C) The solution can be checked by foiling it. If the result of foiling is the same as the original question, then the solution is correct.

e.g. Factor x 2−1 then check the result. x 2−1 = x +1( ) x −1( )

Check: x +1( ) x −1( ) = x 2 +1x −1x −1 = x 2−1 . Since the result of foiling is x 2−1 , which is the same

as the original question, the factors

x +1( ) x −1( ) are correct.



D) REQUIRED PRACTICE 3: Factor these expressions. SHOW THE PROCESS!! {Answers are on page 4 of these notes.} 1) x

2−64 2) y2−100 3) 3x

2−27 4) 4x2−81 5) 4x

2−25

6) 3x2−75 7) 8x

2−2 8) 5x2−20 9) 48a2−147b2 10) 20x 3− 405xy 2

11) 100x 3y 2−324xy 4

III) A SUM OF PERFECT SQUARES IS A PERFECT SQUARE ADDED TO A DIFFERENT PERFECT SQUARE

e.g. x2 +y 2( ); x 2 + 4( ); 9x 2 + 25( ); 4x 2 + 36y 2( ); x 2 +1( )

A) A sum of perfect squares cannot be factored because it has no real factors. Consider the sum of perfect squares: x2 + 4. You may think its factors are (x + 2)(x + 2), however, when these binomials are foiled, the result is the quadratic trinomial x2 + 4x + 4, not the original quadratic binomial x2 + 4. You may think its factors are (x – 2)(x – 2), however, when these quadratic binomial are foiled, the result is the quadratic trinomial x2 – 4x+ 4, not the original quadratic binomial x2 + 4. You may think its factors are (x + 2)(x – 2), however, when these quadratic binomial are foiled, the result is the quadratic binomial x2 – 4, not the original binomial x2 + 4. Since all possible factors do not reproduce the original expression, the original sum of perfect squares cannot be factored.

A SUM OF PERFECT SQUARES CANNOT BE FACTORED!! REMEMBER:

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

1) LEADING COEFFICIENT = the coefficient of the x-variable having the largest exponent.

2) QUADRATIC TRINOMIAL = a quadratic relation written in descending order of x, which is called standard form: ax 2 +bx +c where a ≠ 0 .

FACTORING QUADRATIC TRINOMIALS I) WHEN A QUADRATIC TRINOMIAL IS WRITTEN IN STANDARD FORM, IT IS WRITTEN IN DESCENDING ORDER OF x:

ax 2 +bx +c . e.g. x

2 + 2x +1 ; x 2−2x +1 ; 3x 2−8x −3 ; 10x 2 + 7x −12 ; 4x 2−24x + 36

A) In order to factor quadratic trinomials, you must be able to identify the values of a, b and c. Study the table given below carefully. The values of a, b and c of each quadratic trinomial are indicated to their right. Be sure you memorize and understand how to determine the values of a, b and c in a quadratic trinomial.

Quadratic Trinomial a b c

x2 + 2x +1 1 2 1

−x 2−2x −3

−1 −2

−3

10x 2 + 7x −12 10 7 −12

4x2−24x + 36 4

−24 36

II) In order to factor a quadratic trinomial it must be written in descending order of x. There are two methods of factoring quadratic trinomials that you will learn. The method used depends upon the leading coefficient. If the leading coefficient is 1, use the SHORT METHOD. If the leading coefficient is any number other than 1, use the LONG METHOD.



A) SHORT METHOD: a = 1 USE THESE STEPS TO FACTOR QUADRATIC TRINOMIALS WHEN

1: COMMON FACTOR if possible. 2: Write two sets of brackets with an x at the beginning of each set. 3: Find two numbers that multiply together to equal c that also add together to equal b. 4: Place one of the numbers at the end of the first set of brackets and the other at the end of the second

set of brackets.

1) SAMPLE PROBLEMS 4: Study these examples carefully. Be sure you understand and memorize the process used to complete them. a) Factor x

2 + 2x +1 b) Factor x2−2x +1 c) Factor x

2 + 5x + 6

d) Factor −x 2 + 7x −6 e) Factor 3x

2 + 3x −60 f) Factor 5x 3−15x 2−50x

2) REQUIRED PRACTICE 4: Factor these quadratic trinomials. SHOW THE PROCESS!! {Answers are on page 5 of these notes.} a) x

2−5x + 6 b) x2 + 4x + 3 c) x

2 + 9x +18 d) x2−6x + 5

e) x2−11x + 24 f)

−x 2−2x + 3 g) −x 2− 4x + 5 h) 3x

2−6x −24 i)

−2x 2−10x + 48 j) −x 2 + 9x −20 k) x

3 + 5x 2−14x l) 3x3−15x 2− 42x

ANSWERS TO THE REQUIRED PRACTICE

Required Practice 1 from page 1 1) 4 x −3y( ) 2) 5xy 1−2x( ) 3) 7x

2y 3 xy−7( ) 4) 6xy2 4xy−7( ) 5) 32a2w aw−2( ) 6) 5at 2 2a2−5( )

7) 7gh 13h−7g( ) 8) 3xy 4y + 9x( ) 9) 7x2y 3 7y + 9x( ) 10) 3 ab−2ac−3bc( ) 11) 2x y−2w−3w2( )

12) −11 4a + 7b + 3c( ) 13) 15a x −2y−3w( ) 14) 5ax x 2−2a−3a2( ) 15)

−2a2x 2 a + 2x + 4( )

16) 3xy 2x + 5y−9( ) 17) 9xy2 4x 3−5xy−2y 2( ) 18)

−7m2n 5n2 + 3n + 8( )

Required Practice 2 from page 3 1) x + 8( ) x −8( ) 2) y +10( ) y−10( ) 3) 3 x + 3( ) x −3( ) 4) 2x + 9( ) 2x −9( ) 5) 2x + 5( ) 2x −5( )

6) 3 x + 5( ) x −5( ) 7) 2 2x +1( ) 2x −1( ) 8) 5 x + 2( ) x −2( ) 9) 3 4a + 7b( ) 4a−7b( ) 10) 5x 2x + 9y( ) 2x −9y( )

11) 4xy2 5x + 9y( ) 5x −9y( )



Required Practice 4 from page 4 a) x −2( ) x −3( ) b) x +1( ) x + 3( ) c) x + 3( ) x + 6( ) d) x −1( ) x −5( ) e) x −3( ) x −8( ) f)

−1 x + 3( ) x −1( )

g) −1 x + 5( ) x −1( ) h) 3 x + 2( ) x − 4( ) i)

−2 x + 8( ) x −3( ) j) −1 x − 4( ) x −5( ) k) x x + 7( ) x −2( )

l) 3x x + 2( ) x −7( )



ASSIGNMENT: PRINT THIS INFORMATION ON YOUR OWN GRID PAPER

LAST then FIRST Name T19 – FACTORING REVIEW - 2 Block:

Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.)

REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!!

∞ Answer this question. 1) When factoring an algebraic expression, what type of factoring is always attempted first? (1)

∞ Factor these expressions. REMEMBER to common factor first. If there are no factors state so. 2)

−8x 2y 4z 2−24x 3y5z 3−56xy 3z 3 (4) 3) −39x 6y5z 4 −52x 3y 6z 2−5x 4y 4z (4) 4) 3x

2y 2 +18x 2y−9xz (4)

5) y2−16 (1) 6) 4x

2 +1 (1) 7) 36x 2−64y 2 (3)

8) −12y 2 + 9x + 6 (2) 9) 3x

2−12x −12 (2) 10) x2 + 7x +12 (2)

11) x2−7x +10 (2) 12) 2x

2 +18x + 40 (4) 13) 4x 2−20x −56 (4)

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