For Common Assessment Adding and Subtracting Polynomials Multiplying Polynomials Factoring...
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Transcript of For Common Assessment Adding and Subtracting Polynomials Multiplying Polynomials Factoring...
For Common Assessment
Adding and Subtracting Polynomials
Multiplying Polynomials
Factoring Polynomials
Adding & Subtracting Polynomials
To add or subtract polynomials,
1) Align The Like Terms
2) Add/Subtract The Like Terms*Subtracting is the same as adding the opposite!!
** When adding or subtracting, EXPONENTS STAY THE SAME!!
There are two ways to add and subtract polynomials. You can do it horizontally or vertically.
•Simplify (2z + 5y) + (3z – 2y)
(2z + 5y) + (3z – 2y)
= 2z + 5y + 3z – 2y
= 2z + 3z + 5y – 2y
= 5z + 3y
Horizontal example:
Line up your like terms. 9y – 7x + 15a
+ -3y + 8x – 8a_________________________
Add the following polynomials (9y – 7x + 15a) + (-3y + 8x – 8a)
6y + x + 7a
3a2 + 3ab – b2
+ 4ab + 6b2
_________________________
Add the following polynomials (3a2 + 3ab – b2) + (4ab + 6b2)
3a2 + 7ab + 5b2
Line up your like terms. 4x2 – 2xy + 3y2
+ -3x2 – xy + 2y2
_________________________
x2 - 3xy + 5y2
Add the following polynomials (4x2 – 2xy + 3y2) + (-3x2 – xy + 2y2)
Line up your like terms and add the opposite.
9y – 7x + 15a+ (+ 3y – 8x + 8a)--------------------------------------
Subtract the following polynomials (9y – 7x + 15a) – (-3y +8x – 8a)
12y – 15x + 23a
7a – 10b+ (– 3a – 4b)--------------------------------------
Subtract the following polynomials (7a – 10b) – (3a + 4b)
4a – 14b
4x2 – 2xy + 3y2
+ (+ 3x2 + xy – 2y2)--------------------------------------
7x2 – xy + y2
Subtract the following polynomials (4x2 – 2xy + 3y2) – (-3x2 – xy
+ 2y2)
Subtract (5x2 + 3a2 – 5x) – (2x2 – 5a2 + 7x)
5x2 + 3a2 – 5x
+ (- 2x2 + 5a2 – 7x)
--------------------------------------
3x2 + 8a2 – 12x
Subtract (3x2 + 8x + 4) – (5x2 – 4)
3x2 + 8x + 4
+ (- 5x2 + 4)
--------------------------------------
-2x2 + 8x + 8
(2x3 + 4x2 - 6) – (3x3 + 2x - 2)
(7x3 - 3x + 1) – (x3 - 4x2 - 2)
(2x3 + 4x2 - 6) + (-3x3 + -2x - -2)
= -x3 + 4x2 - 2x - 4
(7x3 - 3x + 1) + (-x3 - -4x2 - -2)
= 6x3 + 4x2 - 3x + 3
Remember that when you multiply two powers with the same bases, you add the exponents.
(5m2n3)(6m3n6)
5 · 6 · m2+3n3+6
30m5n9
Pre-Algebra
To multiply two monomials, multiply the coefficients and add the exponents of the variables that are the same.
MULTIPLYING POLYNOMIALS
Multiply.
Multiplying Monomials
A. (2x3y2)(6x5y3)
(2x3y2)(6x5y3)
12x8y5
Multiply coefficients and addexponents.
B. (9a5b7)(–2a4b3)
(9a5b7)(–2a4b3)
–18a9b10
Multiply coefficients and addexponents.
Pre-Algebra
Try This
Multiply.
A. (5r4s3)(3r3s2)
(5r4s3)(3r3s2)
15r7s5
Multiply coefficients and addexponents.
B. (7x3y5)(–3x3y2)
(7x3y5)(–3x3y2)
–21x6y7
Multiply coefficients and addexponents.
Multiply.
Multiplying a Polynomial by a Monomial
A. 3m(5m2 + 2m)
3m(5m2 + 2m)
15m3 + 6m2
Multiply each term in parentheses by 3m.
B. –6x2y3(5xy4 + 3x4)
–6x2y3(5xy4 + 3x4)
–30x3y7 – 18x6y3
Multiply each term in parentheses by –6x2y3.
Multiply.
Multiplying a Polynomial by a Monomial
C. –5y3(y2 + 6y – 8)
–5y3(y2 + 6y – 8)
–5y5 – 30y4 + 40y3
Multiply each term in parentheses by –5y3.
Pre-Algebra
Try This: Example 2A & 2B
Multiply.
Insert Lesson Title Here
A. 4r(8r3 + 16r)
4r(8r3 + 16r)
32r4 + 64r2
Multiply each term in parentheses by 4r.
B. –3a3b2(4ab3 + 4a2)
–3a3b2(4ab3 + 4a2)
–12a4b5 – 12a5b2
Multiply each term in parentheses by –3a3b2.
Example 2
Insert Lesson Title Here
Multiply.
C. –2x4(x3 + 4x + 3)
–2x4(x3 + 4x + 3)
–2x7 – 8x5 – 6x4
Multiply each term in parentheses by –2x4.
Pre-Algebra
Multiply. (2x + 3)(5x + 8)
Using the Distributive property, multiply
2x(5x + 8) + 3(5x + 8).
10x2 + 16x + 15x + 24
Combine like terms.
10x2 + 31x + 24
Another option is called the FOIL method.
EXAMPLES
( x + 4 ) ( x + 8 ) =
( x + 5 ) ( x – 6) =
x2 + 8x + 4x + 32
x2 − 6x + 5x − 30
x2 + 12x + 32
x2 + 12x + 32
x2 − x − 30 x2 − x − 30
PRACTICE
( x − 7 ) ( x − 4 ) =
( x + 10 ) ( x + 3 ) =
x2 + 3x + 10x + 30
x2 + 13x + 30
x2 + 3x + 10x + 30
x2 + 13x + 30
x2 − 4x − 7x + 28
x2 − 11x + 28
x2 − 4x − 7x + 28
x2 − 11x + 28
( 2x2 + 4 ) ( 3x − 5 ) =
( 3x2 − 6x) (4x + 2) =
EXAMPLES
6x3 − 10x2 + 12x− 20
12x3 + 6x2
− 24x2
− 12x
12x3 − 18x2 − 12x12x3 − 18x2 − 12x
3(8 12)
4a a 2) Simplify:
6a2 + 9a
3) Simplify: 6rs(r2s - 3) 6rs • r2s
6r3s2 - 18rs
38
4a a
312
4a
- 6rs • 3
4) Simplify: 4t2(3t2 + 2t - 5)
12t4
5) Simplify: - 4m3(-3m - 6n + 4p)
12m4
+ 8t3- 20t2
+ 24m3n - 16m3p
Simplify -3x2y3(y2 – x2 + 2xy)
1. -3x2y5 + 3x4y3 – 6x3y4
2. -3x2y6 + 3x4y3 – 6x2y3
3. -3x2y5 + 3x4y3 – 6x2y3
4. 3x2y5 – 3x4y3 + 6x3y4
39
Multiply: – 3x2y(5x2 – 2xy + 7y2).
= – 3x2y(5x2 ) – 3x2y(– 2xy) – 3x2y(7y2)
= – 15x4y + 6x3y2 – 21x2y3
40
Example: Multiply: (x – 1)(2x2 + 7x + 3).= (x – 1)(2x2) + (x – 1)(7x) + (x – 1)(3)= 2x3 – 2x2 + 7x2 – 7x + 3x – 3= 2x3 + 5x2 – 4x – 3
41
Examples: Multiply: (2x + 1)(7x – 5).
= 2x(7x) + 2x(–5) + (1)(7x) + (1)(– 5)
= 14x2 – 10x + 7x – 5
= 14x2 – 3x – 5
First Outer Inner Last
42
Multiply: (5x – 3y)(7x + 6y).
= 35x2 + 30xy – 21yx – 18y2
= 35x2 + 9xy – 18y2
= 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y)First Outer Inner Last
43
(a + b)(a – b)
= a2 – b2
The multiply the sum and difference of two terms, use this pattern:
= a2 – ab + ab – b2
square of the first termsquare of the second term
Special Cases
45
(a + b)2 = (a + b)(a + b)
= a2 + 2ab + b2
= a2 + ab + ab + b2
To square a binomial, use this pattern:
square of the first term
twice the product of the two terms square of the last term
Special Cases
46
Examples: Multiply: (2x – 2)2 .
= (2x)2 + 2(2x)(– 2) + (– 2)2
= 4x2 – 8x + 4
Multiply: (x + 3y)2 .= (x)2 + 2(x)(3y) + (3y)2
= x2 + 6xy + 9y2
Special Cases
Techniques of Factoring Polynomials
1. Greatest Common Factor (GCF). The GCF for a polynomial is the largest monomial that divides each term of the polynomial.
Factor out the GCF: 23 24 yy
Factoring Polynomials - GCF
23 24 yy
y2
yyy22
Write the two terms in the form of prime factors…
They have in common 2yy
)12(2 2 yy
yy2
1)(2yy
This process is basically the reverse of the distributive property.
50
The simplest method of factoring a polynomial is to factor out the greatest common factor (GCF) of each term.
Example: Factor 18x3 + 60x.
GCF = 6x18x3 + 60x = 6x (3x2) + 6x (10) Apply the distributive law
to factor the polynomial.
6x (3x2 + 10) = 6x (3x2) + 6x (10) = 18x3 + 60x
Check the answer by multiplication.
Find the GCF.
= 6x (3x2 + 10)
Factoring - GCF
51
Example: Factor 4x2 – 12x + 20.
Therefore, GCF = 4.
4x2 – 12x + 20 = 4x2 – 4 · 3x + 4 · 5
4(x2 – 3x + 5) = 4x2 – 12x + 20
Check the answer.= 4(x2 – 3x + 5)
Factoring - GCF
52
A common binomial factor can be factored out of certain expressions.
Example: Factor the expression 5(x + 1) – y(x + 1).
5(x + 1) – y(x + 1) = (5 – y)(x + 1)
(5 – y)(x + 1) = 5(x + 1) – y(x + 1)
Check.
Factoring - GCF
Factoring Polynomials - GCF
Factor the GCF:
24233 8124 cabcbaab
3 terms
4ab2( )b - 3a c2 + 2b c2 2
One term
57
To factor a trinomial of the form x2 + bx + c, express the trinomial as the product of two binomials. For example,
x2 + 10x + 24 = (x + 4)(x + 6).
Factoring – Sum and Product
4 and 6 add up to 10
4 and 6 multiply to 24
61
65
67
2
2
2
xx
xx
xx
factors of 6 that add up to 7: 6 and 1
1)6( xx
factors of – 6 that add up to – 5: – 6 and 1
factors of – 6 that add up to 1: 3 and – 2
1)6( xx
2)3( xx
Factoring Trinomials
59
Example: Factor x2 – 8x + 15 = (x + a)(x + b)
x2 – 8x + 15 = (x – 3)(x – 5).
Therefore a + b = – 8
= x2 + (a + b)x + ab
It follows that both a and b are negative.
and ab = 15.
Factoring – Sum and Product
60
Example: Factor x2 + 13x + 36. = (x + a)(x + b)
Therefore a and b are two positive factors of 36 whose sum is 13.
x2 + 13x + 36 = (x + 4)(x + 9)
= x2 + (a + b) x + ab
Factoring – Sum and Product
)2()3(
)3(2)3(
623
2
2
23
xx
xxx
xxx There is no GCF for allfour terms.
In this problem we factor GCFby grouping the first two terms and the last two terms.
Factoring 4 Terms by Grouping
62
Some polynomials can be factored by grouping terms to produce a common binomial factor.
= (2x + 3)y – (2x + 3)2
= (2xy + 3y) – (4x + 6) Group terms.
Examples: Factor 2xy + 3y – 4x – 6.
Factor each pair of terms.= (2x + 3)( y – 2) Factor out the common binomial.
Factoring – By Grouping 4 Terms
63
Factor 2a2 + 3bc – 2ab – 3ac.
= 2a2 – 2ab + 3bc – 3ac = (2a2 – 2ab) + (3bc – 3ac)
= 2a(a – b) + 3c(b – a)
Rearrange terms.
Group terms.
Factor.
= 2a(a – b) – 3c(a – b) b – a = – (a – b).
= (2a – 3c)(a – b) Factor.
2a2 + 3bc – 2ab – 3ac
Factoring – By Grouping 4 Terms
Factoring a trinomial when a ≠ 1
Factor 8b2 + 2b – 3
Multiply 8 -3, and break up the middle term
€
8b2 + 6b − 4b − 3 Now factor by grouping
€
8b2 + 6b( ) −4b − 3( )
€
2b(4b + 3) −1(4b + 3)
€
(4b + 3)(2b −1)
Factoring a trinomial when a ≠ 1
Factor 2x2 + 19x - 10 Multiply 8 -3, and break up the middle term
€
8b2 + 6b − 4b − 3 Now factor by grouping
€
8b2 + 6b( ) −4b − 3( )
€
2b(4b + 3) −1(4b + 3)
€
(4b + 3)(2b −1)
Factoring a trinomial when a ≠ 1
Factor 6y2 – 11y - 10 Multiply 8 -3, and break up the middle term
€
8b2 + 6b − 4b − 3 Now factor by grouping
€
8b2 + 6b( ) −4b − 3( )
€
2b(4b + 3) −1(4b + 3)
€
(4b + 3)(2b −1)
Factoring a trinomial when a ≠ 1
Factor 2x2 – x – 3 Multiply 8 -3, and break up the middle term
€
8b2 + 6b − 4b − 3 Now factor by grouping
€
8b2 + 6b( ) −4b − 3( )
€
2b(4b + 3) −1(4b + 3)
€
(4b + 3)(2b −1)
Factoring a trinomial when a ≠ 1
Factor 3t2 + 16t + 5 Multiply 8 -3, and break up the middle term
€
8b2 + 6b − 4b − 3 Now factor by grouping
€
8b2 + 6b( ) −4b − 3( )
€
2b(4b + 3) −1(4b + 3)
€
(4b + 3)(2b −1)
Factoring a trinomial when a ≠ 1
Factor 5x2 + 2x – 3 Multiply 8 -3, and break up the middle term
€
8b2 + 6b − 4b − 3 Now factor by grouping
€
8b2 + 6b( ) −4b − 3( )
€
2b(4b + 3) −1(4b + 3)
€
(4b + 3)(2b −1)
Factoring a trinomial when a ≠ 1
Factor 6b2 – 11b – 2 Multiply 8 -3, and break up the middle term
€
8b2 + 6b − 4b − 3 Now factor by grouping
€
8b2 + 6b( ) −4b − 3( )
€
2b(4b + 3) −1(4b + 3)
€
(4b + 3)(2b −1)
Factoring the Difference of Two Squares
The difference of two bases being squared, factors as the product of the sum and difference of the bases that are being squared.
a2 – b2 = (a + b)(a – b) FORMULA:
(a + b)(a – b) = a2– ab + ab – b2 = a2 – b2
Factoring the difference of two squares
Factor x2 – 4y2 Factor 16r2 – 25
(x)2 (2y)2
(x – 2y)(x + 2y)
(4r)2 (5)2
Difference of two squares
DifferenceOf two squares
(4r – 5)(4r + 5)
a2 – b2 = (a + b)(a – b)
76
A difference of squares can be factored
using the formula
Example: Factor x2 – 9y2.
= (x)2 – (3y)2
= (x + 3y)(x – 3y)
Write terms as perfect squares.
a2 – b2 = (a + b)(a – b).
Factoring – Special Products