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

3a 2 + 3ab – b 2 + 4ab + 6b 2 _________________________ Add the following polynomials (3a 2 + 3ab – b 2 ) + (4ab + 6b 2 ) 3a 2 + 7ab+ 5b 2

Line up your like terms. 4x 2 – 2xy + 3y 2 +-3x 2 – xy + 2y 2 _________________________ x 2 - 3xy + 5y 2 Add the following polynomials (4x 2 – 2xy + 3y 2 ) + (-3x 2 – xy + 2y 2 )

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

4x 2 – 2xy + 3y 2 + (+ 3x 2 + xy – 2y 2 ) x 2 – xy + y 2 Subtract the following polynomials (4x 2 – 2xy + 3y 2 ) – (-3x 2 – xy + 2y 2 )

Subtract (5x 2 + 3a 2 – 5x) – (2x 2 – 5a 2 + 7x) 5x 2 + 3a 2 – 5x + (- 2x 2 + 5a 2 – 7x) x 2 + 8a 2 – 12x

Subtract (3x 2 + 8x + 4) – (5x 2 – 4) 3x 2 + 8x (- 5x 2 + 4) x 2 + 8x + 8

Find the sum or difference. (5a – 3b) + (2a + 6b) 1.3a – 9b 2.3a + 3b 3.7a + 3b 4.7a – 3b

Find the sum or difference. (5a – 3b) – (2a + 6b) 1.3a – 9b 2.3a + 3b 3.7a + 3b 4.7a – 9b

(5x 2 - 3x + 7) + (2x 2 + 5x - 7) = 7x 2 + 2x (3x 3 + 6x - 8) + (4x 2 + 2x - 5) = 3x 3 + 4x 2 + 8x - 13

(2x 3 + 4x 2 - 6) – (3x 3 + 2x - 2) (7x 3 - 3x + 1) – (x 3 - 4x 2 - 2) (2x 3 + 4x 2 - 6) + (-3x x - -2) = -x 3 + 4x 2 - 2x - 4 (7x 3 - 3x + 1) + (-x x ) = 6x 3 + 4x 2 - 3x + 3

7y 2 – 3y y 2 + 3y – 4 2x 3 – 5x 2 + 3x – 1 – (8x 3 – 8x 2 + 4x + 3) –6x 3 + 3x 2 – x – 4 = 15y 2

(7y 3 +2y 2 + 5y – 1) + (5y 3 + 7y) 12y 3 + 2y y – 1

(b 4 – 6 + 5b + 1) + (8b 4 + 2b – 3b 2 ) = 9b 4 – 3b 2 + 7b – 5

Remember that when you multiply two powers with the same bases, you add the exponents. (5m 2 n 3 )(6m 3 n 6 ) 5 · 6 · m 2+3 n m 5 n 9 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. (2x 3 y 2 )(6x 5 y 3 ) (2x 3 y 2 )(6x 5 y 3 ) 12x 8 y 5 Multiply coefficients and add exponents. B. (9a 5 b 7 )( – 2a 4 b 3 ) (9a 5 b 7 )( – 2a 4 b 3 ) – 18a 9 b 10 Multiply coefficients and add exponents. Pre-Algebra

Try This Multiply. A. (5r 4 s 3 )(3r 3 s 2 ) (5r 4 s 3 )(3r 3 s 2 ) 15r 7 s 5 Multiply coefficients and add exponents. B. (7x 3 y 5 )( – 3x 3 y 2 ) (7x 3 y 5 )( – 3x 3 y 2 ) – 21x 6 y 7 Multiply coefficients and add exponents.

Multiply. Multiplying a Polynomial by a Monomial A. 3m(5m 2 + 2m) 3m(5m 2 + 2m) 15m 3 + 6m 2 Multiply each term in parentheses by 3m. B. – 6x 2 y 3 (5xy 4 + 3x 4 ) – 6x 2 y 3 (5xy 4 + 3x 4 ) – 30x 3 y 7 – 18x 6 y 3 Multiply each term in parentheses by – 6x 2 y 3.

Multiply. Multiplying a Polynomial by a Monomial C. – 5y 3 (y 2 + 6y – 8) – 5y 3 (y 2 + 6y – 8) – 5y 5 – 30y y 3 Multiply each term in parentheses by – 5y 3. Pre-Algebra

Try This: Example 2A & 2B Multiply. Insert Lesson Title Here A. 4r(8r r) 4r(8r r) 32r r 2 Multiply each term in parentheses by 4r. B. – 3a 3 b 2 (4ab 3 + 4a 2 ) – 3a 3 b 2 (4ab 3 + 4a 2 ) – 12a 4 b 5 – 12a 5 b 2 Multiply each term in parentheses by – 3a 3 b 2.

Multiply. (2x + 3)(5x + 8) Using the Distributive property, multiply 2x(5x + 8) + 3(5x + 8). 10x x + 15x + 24 Combine like terms. 10x x + 24 Another option is called the FOIL method.

EXAMPLES ( x + 4 ) ( x + 8 ) = ( x + 5 ) ( x – 6) = x2x2 + 8x+ 4x+ 32 x2x2 − 6x+ 5x− 30 x x + 32 x x + 32 x 2 − x − 30

PRACTICE ( x − 7 ) ( x − 4 ) = ( x + 10 ) ( x + 3 ) = x 2 + 3x + 10x + 30 x x + 30 x 2 + 3x + 10x + 30 x x + 30 x 2 − 4x − 7x + 28 x 2 − 11x + 28 x 2 − 4x − 7x + 28 x 2 − 11x + 28

( 2x ) ( 3x − 5 ) = ( 3x 2 − 6x) (4x + 2) = EXAMPLES 6x 3 − 10x x− 20 12x 3 + 6x 2 − 24x 2 − 12x 12x 3 − 18x 2 − 12x

Example: (x +3)(x+1)=(x)(x)+(x)(1)+(3)(x)+(3)((1)

1) Simplify: 5(7n - 2) Use the distributive property. 5 7n 35n

2) Simplify: 6a 2 + 9a 3) Simplify: 6rs(r 2 s - 3) 6rs r 2 s 6r 3 s rs - 6rs 3

4) Simplify: 4t 2 (3t 2 + 2t - 5) 12t 4 5) Simplify: - 4m 3 (-3m - 6n + 4p) 12m 4 + 8t t m 3 n- 16m 3 p

Simplify 4y(3y 2 – 1) 1.7y 2 – y 2 – y 3 – y 3 – 4y

Simplify -3x 2 y 3 (y 2 – x 2 + 2xy) 1.-3x 2 y 5 + 3x 4 y 3 – 6x 3 y x 2 y 6 + 3x 4 y 3 – 6x 2 y x 2 y 5 + 3x 4 y 3 – 6x 2 y 3 4.3x 2 y 5 – 3x 4 y 3 + 6x 3 y 4

Try These. 1.) (x+2) (x+8) = x 2 +10x+16 2.) (x+5) (x-7) = x 2 -2x-35 3.) (2x+4) (2x-3) = 4x 2 +2x-12

36 Examples:. Multiply: 2x(3x 2 + 2x – 1). = 6x 3 + 4x 2 – 2x = 2x(3x 2 ) + 2x(2x) + 2x(–1)

37 Multiply: – 3x 2 y(5x 2 – 2xy + 7y 2 ). = – 3x 2 y(5x 2 ) – 3x 2 y(– 2xy) – 3x 2 y(7y 2 ) = – 15x 4 y + 6x 3 y 2 – 21x 2 y 3

38 Example: Multiply: (x – 1)(2x 2 + 7x + 3). = (x – 1)(2x 2 ) + (x – 1)(7x) + (x – 1)(3) = 2x 3 – 2x 2 + 7x 2 – 7x + 3x – 3 = 2x 3 + 5x 2 – 4x – 3

39 Examples: Multiply: (2x + 1)(7x – 5). = 2x(7x) + 2x(–5) + (1)(7x) + (1)(– 5) = 14x 2 – 10x + 7x – 5 = 14x 2 – 3x – 5 First Outer Inner Last

40 Multiply: (5x – 3y)(7x + 6y). = 35x xy – 21yx – 18y 2 = 35x 2 + 9xy – 18y 2 = 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y) First Outer Inner Last

41 (a + b)(a – b) = a 2 – b 2 The multiply the sum and difference of two terms, use this pattern: = a 2 – ab + ab – b 2 square of the first term square of the second term Special Cases

42 Examples: (3x + 2)(3x – 2) = (3x) 2 – (2) 2 = 9x 2 – 4 (x + 1)(x – 1) = (x) 2 – (1) 2 = x 2 – 1

43 (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2 = a 2 + ab + ab + b 2 To square a binomial, use this pattern: square of the first term twice the product of the two termssquare of the last term Special Cases

44 Examples: Multiply: (2x – 2) 2. = (2x) 2 + 2(2x)(– 2) + (– 2) 2 = 4x 2 – 8x + 4 Multiply: (x + 3y) 2. = (x) 2 + 2(x)(3y) + (3y) 2 = x 2 + 6xy + 9y 2 Special Cases

FACTORING GCF Method Sum + Product Method Factor by Grouping 4 Terms Method

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:

Factoring Polynomials - GCF Write the two terms in the form of prime factors… They have in common 2yy This process is basically the reverse of the distributive property.

48 The simplest method of factoring a polynomial is to factor out the greatest common factor (GCF) of each term. Example: Factor 18x x. GCF = 6x 18x x = 6x (3x 2 ) + 6x (10) Apply the distributive law to factor the polynomial. 6x (3x ) = 6x (3x 2 ) + 6x (10) = 18x x Check the answer by multiplication. Find the GCF. = 6x (3x ) Factoring - GCF

49 Example: Factor 4x 2 – 12x Therefore, GCF = 4. 4x 2 – 12x + 20 = 4x 2 – 4 · 3x + 4 · 5 4(x 2 – 3x + 5) = 4x 2 – 12x + 20 Check the answer. = 4(x 2 – 3x + 5) Factoring - GCF

50 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: 3 terms 4 a b 2 ( )b - 3a c 2 + 2b c 22 One term

Factoring Polynomials - GCF EXAMPLE: 5x- 3

Examples Factor the following polynomial.

55 To factor a trinomial of the form x 2 + bx + c, express the trinomial as the product of two binomials. For example, x x + 24 = (x + 4)(x + 6). Factoring – Sum and Product 4 and 6 add up to 10 4 and 6 multiply to 24

factors of 6 that add up to 7: 6 and 1 factors of – 6 that add up to – 5: – 6 and 1 factors of – 6 that add up to 1: 3 and – 2 Factoring Trinomials

57 Example: Factor x 2 – 8x + 15 = (x + a)(x + b) x 2 – 8x + 15 = (x – 3)(x – 5). Therefore a + b = – 8 = x 2 + (a + b)x + ab It follows that both a and b are negative. and ab = 15. Factoring – Sum and Product

58 Example: Factor x x = (x + a)(x + b) Therefore a and b are two positive factors of 36 whose sum is 13. x x + 36= (x + 4)(x + 9) = x 2 + (a + b) x + ab Factoring – Sum and Product

There is no GCF for all four terms. In this problem we factor GCF by grouping the first two terms and the last two terms. Factoring 4 Terms by Grouping

60 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

61 Factor 2a 2 + 3bc – 2ab – 3ac. = 2a 2 – 2ab + 3bc – 3ac = (2a 2 – 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. 2a 2 + 3bc – 2ab – 3ac Factoring – By Grouping 4 Terms