Adding and Subtracting Polynomials

Section 0.3

Polynomial

A polynomial in x is an algebraic expression of the form:

The degree of the polynomial is n (largest exponent) The leading coefficient is ( the coefficient on term

with highest exponent) The constant term is (the term without a

variable) The polynomial should be written in standard form.

(Decreasing order according to exponents)

a x a x a x a x a x ann

nn

nn

1

12

22

21 0...

an

a0

Polynomials

91523 34 xxx

Leading Coefficient:

Degree:

Constant:

4

3

-9

Polynomials

Naming a polynomial: 1 term - monomial 2 terms - binomial 3 terms - trinomial 4 or more - terms polynomial

Example 2x + 7 has 2 terms so it is called a binomial

Classifying Polynomials

(a) 2 t + 7 4 The polynomial cannot be simplified.

The degree is 4.The polynomial is a binomial.

The polynomial can be simplified.

The degree is 2.The simplified polynomial is

a monomial.

(b) 3 e + 5 e – 9 e2 2 2

= – e 2

Two terms.

One term.

Combine like terms and put the polynomial in standard form. What degree is the polynomial? Name the polynomial by the number of terms.

5 3 74 4 5x x x x

8 74 5x x x

7 85 4x x x

Degree is 5

Trinomial

Adding Polynomials

Adding Polynomials Horizontally

Add 2n – 7n – 4 and – 5n – 8n + 10.4 3 4 3

( 2n – 7n – 4 ) + ( – 5n – 8n + 10 )4 3 4 3

– 3n4 – 15n3 + 6=

Find the sum (8y – 7y – y + 3) + (6y + 2y – 4y + 1).3 2 3 2

+ 4– 5y 14y3 2 – 5y

Adding Polynomials

Subtracting Polynomials

Subtracting PolynomialsTo subtract two polynomials, change all the signs of the secondpolynomial and add the result to the first polynomial.

(Distribute the negative)

Subtracting Polynomials

Perform the subtraction ( 3x – 5 ) – ( 6x – 4 ).

( 3x – 5 ) – ( 6x – 4 )

Change the signs in the second polynomial.

– 3x = – 1

= 3x – 5 – 6x + 4

Subtracting Multivariable Polynomials

Add or subtract as indicated.

– ab

( 2a b – 4ab + b ) – ( 5a b – 3ab + 7b )2 2 2 2

= 2a b – 4ab + b – 5a b + 3ab – 7b 2 2 2 2

= – 3a b2 – 6b2

Multiplying Polynomials

(a) 5x ( 6x + 7 ) 2 4

Distributive property= 5x ( 6x ) 2 4 + 5x ( 7 ) 2

= 30x + 35x 6 2 Multiply monomials.

Use the distributive property to find each product.

Multiplying Polynomials

(b) – 2h ( – 3h + 8h – 1 ) 4 9 2

Use the distributive property to find each product.

136h 616h 42h

Multiplying Binomial times Binomial

F

( 3g + 2 ) ( 9g – 4 )

O

I

L

3g ( 9g )Multiply the First terms:

3g ( – 4 )Multiply the Outer terms:

2 ( 9g )Multiply the Inner terms:

2 ( – 4 )Multiply the Last terms:

= 27g – 12g + 18g – 8 2

= 27g + 6g – 82

F O I L

Multiplying Polynomials

( 6a + 3b ) ( 4a – 2b )

= 24a – 6b2 2

224a ab12 ab12 26b

Multiplying Binomial times Trinomial (Megafoil)

Distributive property

Multiply ( 2y – 5 )( 2y – 7y + 4 ).2 3

( 2y – 5 )( 2y – 7y + 4 )2 3

= (2y )2 (2y )3 (–7y)(2y )2+ (4)(2y )2+

(–7y)+ (–5) (4)+ (–5)(–5)(2y )3+

= 4y5 14y 3– 8y2+ – 20+ 35y– 10y3

= 4y5 24y 3– 8y2+ + 35y – 20 Combine like terms.

Square a binomial

(x+4)²(x+4)(x+4)x² + 4x + 4x + 16x² + 8x + 16

(x-7)²(x-7)(x-7)x² - 7x - 7x + 49x² - 14x + 49

Square the binomial

(2a-3b)²

(2a-3b)(2a-3b)

4a² - 6ab - 6ab + 9b²

4a² - 12ab + 9b²

Find the product

(x+7)(x-7)

x² - 7x + 7x – 49

x² - 49

(2x - ½)(2x + ½)

4x² + x – x - ¼

4x² - ¼

Simplify as much as possible

-(2x – 6)²

-(2x – 6) (2x – 6)

-(4x² - 12x – 12x + 36)

-(4x² - 24x + 36)

-4x² + 24x – 36

3(2x – 4y)²

3(2x – 4y) (2x – 4y)

3(4x² - 8xy – 8xy + 16y²)

3(4x² - 16xy + 16y²)

12x² - 48xy + 48y²

Cubing a Binomial

(x + 4)³

= (x + 4) (x + 4) (x + 4)

= (x + 4)(x² + 8x + 16)= x(x²) + x(8x) + x(16) + 4(x²) + 4(8x) + 4(16)

= x³ + 8x² + 16x + 4x² + 32x + 64

= x³ + 12x² + 48x + 64

Cubing a Binomial

(2x – 3)³

(2x – 3)(2x – 3)(2x – 3)

(2x – 3)(4x² - 12x + 9)2x(4x²) + 2x(-12x) + 2x(9) – 3(4x²) – 3(-12x) – 3(9)

8x³ - 24x² + 18x – 12x² + 36x – 27

8x³ - 36x² + 54x – 27