Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Functional decomposition wikipedia , lookup
Big O notation wikipedia , lookup
Law of large numbers wikipedia , lookup
Mathematical model wikipedia , lookup
Horner's method wikipedia , lookup
Vincent's theorem wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
Adding, Subtracting, and Multiplying Polynomials Monomial: An expression that is a number, a variable, or numbers and variables multiplied together. Monomials only have variables with whole number exponents and never have variables in the denominator of a fraction or variables under roots. xyz 1 , − w, 23, x 2 , x3 y 4 Monomials: 5b, 8 3 1 Not Monomials: 4 , x 3 −1 x, a , z 1 5 Constant: A monomial that contains no variables, like 23 or −1. Coefficient: The numerical part of a monomial (the number being multiplied by the variables.) Polynomial: A monomial or several monomials joined by + or – signs. Terms: The monomials that make up a polynomial. Terms are separated by + or − signs. Like Terms: Terms whose variables and exponents are exactly the same. Binomial: A polynomial with two unlike terms. Trinomial: A polynomial with three unlike terms. Examples: Decide whether each expression is a polynomial. If it isn’t, explain why not. 4 12 a) 5 x 4 + 2 x 3 + 6 x b) − a − a 5 c) d) 6c −2 + c − 1 3 m+2 1 e) 6 z 2 + 5 z 2 − 2 f) 7 g) −8n − 3 h) 3 x + 2 Adding and Subtracting Polynomials To add or subtract polynomials, combine like terms. Add or subtract the coefficients. The variables and exponents do not change. Remember to subtract everything inside the parentheses after a minus sign. Subtract means “add the opposite,” so change the minus sign to a plus sign and then change the signs of all the terms inside the parentheses. Examples: Simplify each expression. a) ( 5n 2 − 2 ) + ( 7 − 3n 2 ) b) ( 2r 2 + 5r ) + ( r 2 − 4r ) c) ( 4 x 2 − 3 x + 1) + ( −2 x 2 + 5 x − 6 ) d) ( 7 z 2 + 12 z − 5 ) + ( 6 z − 4 z 2 − 3) e) ( 2 w2 + 3w ) − ( 4 w2 + w ) f) ( u 3 − 4u 2 + u ) − ( 2u 2 − 5u 3 ) g) ( −6 x 2 − 3 x + 2 ) − ( −4 x 2 − x + 3) h) ( 4 y 2 + 12 y − 7 ) − ( 20 y + 5 y 2 − 8 ) i) ( 6m 2 + 5m ) − ( 4m 2 − 2m ) + ( 3m 2 − 7 m ) j) ( −2k + 5 ) + ( k 2 − 3k ) − ( −4k 2 + 8 ) Multiplying Polynomials To multiply two polynomials, multiply each term of one polynomial by each term of the other polynomial. Then combine any like terms. When you are multiplying two binomials, this is sometimes called the FOIL Method because you multiply F the first terms, O the outside terms, I the inside terms, and L the last terms. Examples: Multiply. a) − xy ( 7 x 2 y + 3 xy − 11) b) ( m + 3)( m − 8 ) c) ( 3 x + 1)( 5 x − 2 ) d) ( −2 z + 5 )( −5 z − 8) e) ( t 2 − 4 ) ( 2t + 9 ) f) ( 2u 2 − 1)( −5u 2 + 4 ) g) ( z + 5 ) 2 i) ( n + 3)( n − 3) k) ( 2 x − 3) ( 5 x 2 − 6 x + 7 ) h) ( 2 x − 3) 2 j) ( 5 y − 2 )( 5 y + 2 ) l) ( 4x 2 + 7 x − 3)( x 2 − 2 x + 8 )