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
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=
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 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
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
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