Unit 3 Polynomials Study Guide

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1 Unit Polynomials Study Guide 7-5 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial Trinomial 4 or more Polynomial Monomial: any number, variable, or product of numbers and variables with whole number exponents. x Examples: 5, x,, 4 x y 4 Binomial: the sum or difference of two unlike monomials Examples: 4x + 5y, 1xy 1x y, 4x 5 Trinomial: the sum or difference of three unlike monomials Examples: 4x + x y+ 5y, x + 5x Polynomial: the sum or difference of four or more unlike monomials Examples: x x + x 1, g g + h h +

2 Part : Classifying Polynomials by Degree Some polynomials have specific names based upon the highest degree. The degree of a monomial is the sum of the exponents on the variables in that monomial. For example: 6 x y z : The degree of the monomial is 6, because = 6 Sum of Exponents Degree of Polynomial Example 0 Constant 5,, 1 Linear x, 4x, 6y, m Quadratic x, 4y, mn Cubic x y, 4m, mnr 4 Quartic 5 Quintic 4 x, 5 x, xy z, x y x y+ x y 6 or more 6 th degree, 7 th degree, 8 th degree xy z, x + x + Part : Writing Polynomials in Standard Form A polynomial is written in Standard Form when the terms are in order of degree from greatest to least. If the degrees add up to the same amount, then arrange the terms alphabetically. Example: 5 6x 7x + 4x + 9 degree: In Standard Form: 7x + 4x + 6x+ 9 ***REMEMBER*** ALWAYS TAKE THE SIGN WITH THE TERM!!!

3 Part 4: Identifying Leading Coefficients of a Polynomial A leading coefficient is the first coefficient of the polynomial when written in Standard Form. Exercises: Example: 6 8 y + y y The leading coefficient is y + y y Find the degree of the polynomial: z 6 4z+ 1. 5k 5k Classify the polynomial by degree and term: 4. s v 4 4 v st + 8s t Write each polynomial in standard form, then identify the leading coefficient: 7. n n h + h a a a + a 7-6 Adding and Subtracting Polynomials Part 1: Adding Polynomials Like Terms: monomials with the same variables raised to the same power. Examples of like terms: x and -x 4x y and -x y When adding or subtracting like terms, the only thing that will change is the COEFFICIENT

4 Examples: #1: (x 5x+ + ( x 8) Step 1: Line up like terms vertically. Remember to keep the sign with the term when you move it. (x 5x+ + ( x Step : Add vertically. (x + ( x x 8) 5x+ 8) 5x 4 # ( 4xy+ x y) (xy+ x 5y) Step 1: Distribute the negative (minus) sign onto the second expression and rewrite. 4 xy+ x y xy x+ 5y Step : Regroup like terms and add or subtract them. 4xy xy+ x x y+ 5y = 7xy+ x+ y Exercises: Add or subtract. Write your answers in standard form. 10. ( y + 5y 6y) + ( 5y 4y+ 1) pr + 6 p 1 7 p+

5 1. ( m m m ) ( m m ) 1. 8pr + 6 p 1 ( 7 p+ ) 7.7 Multiplying Polynomials To multiply monomials: 1. multiply the coefficients. multiply the variables with like bases add the exponents To multiply a polynomial by a monomial: 1. distribute the monomial to each term in the polynomial. multiply the coefficients. multiply the variables with like bases add the exponents Examples: #1 ( 5x y )( xy) ( 5)( )( x x)( y y) 4 10x y # x ( x + 5x+ x ( x ) x( 5x) + x( + 4x + 10x + 8x

6 To multiply binomials: Method 1: Lattice Method Given: ( x )( x+ 1. Since there are four terms in the problem, draw a box and split it into quarters. Write one binomial on the top of the box and one on the side as shown below. x +4 x -. Multiply the terms and fill in the boxes. x - x x -6x +4 +4x -1. Re-write the polynomial in standard form and simplify by combining like terms. x 6x+ 4x 1 x x 1 Method : Vertical Method Given: ( x + )( x 5) 1. Line up the binomials as if you were multiplying two two-digit numbers. x+ x 5. Multiply as if you had a ones digit and a tens digit. Simplify by adding like terms. x+ x 5 15x 10 x + x x 1x 10

7 Exercises: Method : F O I L FOIL is an algorithm mathematical process that stands for: F = First multiply the first terms in each binomial O = Outside multiply the outside terms in each binomial I = Inside multiply the inside terms in each binomial L = Last multiply the last terms in each binomial. Given: ( x + 1)( x+ ) F: -x(-x)= x O: -x() = -4x I: 1(-x) = -x L: 1() = Simplify by writing in standard form and combining like terms. x 5x+ Multiply. Write your answers in standard form. 14. ( y )( 5y 4y+ 1) 15. ( xyz)( 4x yz ) 16. 4x ( x + 8x )

8 17. ( x 5)( x 18. ( x )( x 6x+ 8) 7.8 Special Products of Binomials ***The methods taught in this section are shortcuts for multiplying binomials.*** You can still use any method for multiplying binomials if you like. Perfect-Square Trinomial: a trinomial that is the product of squaring a binomial. Algebraic Definition: ( a + b) = ( a+ b)( a+ b) = a + ab+ b ( a b) = ( a b)( a b) = a ab+ b Examples: #1 ( x + Step 1: identify a and b a = x b = 4 Step : substitute the values of a and b into the above algebraic definition ( x ) + ( x)( + ( Step : simplify x + 8x+ 16

9 # ( x ) Step 1: identify a and b a = x b = - Step : substitute the values of a and b into the above algebraic definition ( x ) + (x)( ) + ( ) Step : simplify 4x 1x+ 9 Difference of Two Squares: a binomial that is the product of multiplying two binomials with like terms but opposite signs. You ll notice that both terms in the product binomial are perfect squares and you are subtracting them Hence the name Difference of Two Squares!! Examples: Algebraic Definition: ( a+ b ) #1 ( x + ( x = a( a b) + b( a b) = a = a ( a b) ab+ ab b b Step 1: identify a and b a = x b = 4 Step : substitute the values of a and b into the above algebraic definition x ( x + 4( x Step : simplify x = x 4x+ 4x 16 16

10 # ( x + ) (x ) Step 1: identify a and b a = x b = Step : substitute the values of a and b into the above algebraic definition x (x ) + (x ) Step : simplify 4x = 4x 4x+ 4x 4 4 Exercises: Multiply. Write your answers in standard form. 19. ( z )( z+ ) 0. ( 5 x + 6) 1. ( 10 + x)( 10 x). ( 7x 5)

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

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