Adding and Subtracting
Polynomials
Section 0.3
Polynomial

A polynomial in x is an algebraic expression of the
form:
an x n  an1x n1  an2 x n2 ...a2 x 2  a1x  a0




The degree of the polynomial is n (largest exponent)
The leading coefficient is an ( the coefficient on term
with highest exponent)
The constant term is a0 (the term without a variable)
The polynomial should be written in standard form.
(Decreasing order according to exponents)
Polynomials
3x 4  2 x 3  15x  9
Degree: 4
Leading Coefficient:
Constant: -9
3
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 4 + 7
Two terms.
(b) 3 e 2 + 5 e 2 – 9 e 2
= –e2
One term.
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.
Combine like terms and put the polynomial in standard
form. What degree is the polynomial? Name the
polynomial by the number of terms.
5x  x  3 x  7 x
4
8x  x  7 x
4
7 x5  8x 4  x
Degree is 5
Trinomial
4
5
5
Adding Polynomials
Adding Polynomials Horizontally
Add 2n 4 – 7n 3 – 4
and
– 5n 4 – 8n 3 + 10.
( 2n 4 – 7n 3 – 4 ) + ( – 5n 4 – 8n 3 + 10 )
=
– 3n4 – 15n 3 + 6
Adding Polynomials
3
2
3
2
Find the sum (8y – 7y – y + 3) + (6y + 2y – 4y + 1).
14y 3 – 5y 2 – 5y + 4
Subtracting Polynomials
Subtracting Polynomials
To subtract two polynomials, change all the signs of the second
polynomial and add the result to the first polynomial.
(Distribute the negative)
Subtracting Polynomials
Perform the subtraction ( 3x – 5 ) – ( 6x – 4 ).
( 3x – 5 ) – ( 6x – 4 )
= 3x – 5 – 6x + 4
= – 3x – 1
Change the signs in the second polynomial.
Subtracting Multivariable
Polynomials
Add or subtract as indicated.
( 2a2 b – 4ab + b 2 ) – ( 5a 2 b – 3ab + 7b 2 )
= 2a 2b – 4ab + b 2 – 5a 2 b + 3ab – 7b 2
= – 3a 2 b – ab – 6b2
Multiplying Polynomials
Use the distributive property to find each product.
(a)
5x2 ( 6x 4 + 7 )
= 5x 2 ( 6x 4) +
=
30x6 +
5x 2 ( 7 )
35x2
Distributive property
Multiply monomials.
Multiplying Polynomials
Use the distributive property to find each product.
(b)
– 2h4 ( – 3h9 + 8h 2 – 1 )
 6h13  16h 6  2h 4
Multiplying Binomial times Binomial
F
( 3g + 2 ) ( 9g – 4 )
O
I
L
= 27g 2 – 12g + 18g – 8
= 27g 2 + 6g – 8
Multiply the First terms:
3g ( 9g )
F
Multiply the Outer terms:
3g ( – 4 )
O
Multiply the Inner terms:
2 ( 9g )
I
Multiply the Last terms:
2(–4)
L
Multiplying Polynomials
( 6a + 3b ) ( 4a – 2b )
 24a  12 ab  12 ab  6b
2
= 24a2 – 6b2
2
Multiplying Binomial times Trinomial
(Megafoil)
Multiply ( 2y 2 – 5 )( 2y 3 – 7y + 4 ).
( 2y 2 – 5 )( 2y 3 – 7y + 4 )
Distributive property
= (2y 2)(2y3 ) + (2y2 ) (–7y) + (2y 2)(4)
+ (–5)(2y3 ) + (–5)(–7y) + (–5)(4)
= 4y 5 – 14y 3 + 8y2 – 10y 3 + 35y – 20
= 4y 5 – 24y 3 + 8y2 + 35y – 20
Combine like terms.
Square a binomial
(x+4)²
(x+4)(x+4)
x² + 4x + 4x + 16
x² + 8x + 16
(x-7)²
(x-7)(x-7)
x² - 7x - 7x + 49
x² - 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