#### Transcript Section 5.1 Polynomials Addition And Subtraction

Section 5.1 Polynomials Addition And Subtraction OBJECTIVES A Classify polynomials. OBJECTIVES B Find the degree of a polynomial and write descending order. OBJECTIVES C Evaluate a polynomial. OBJECTIVES D Add or subtract polynomials. OBJECTIVES E Solve applications involving sums or differences of polynomials. DEFINITION Degree of a Polynomial in One Variable The degree of a polynomial in one variable is the greatest exponent of that variable. DEFINITION Degree of a Polynomial in Several Variables The greatest sum of the exponents of the variables in any one term of the polynomial. RULES Properties for Adding Polynomials If P, Q, and R are polynomials, P +Q =Q + P Commutative Property RULES Properties for Adding Polynomials If P, Q, and R are polynomials, P + Q + R = P + Q + R Associative Property RULES Properties for Adding Polynomials If P, Q, and R are polynomials, P Q + R = PQ + PR Q + R P = QP + RP RULES Subtracting Polynomials a – (b + c) = a – b – c Chapter 5 Section 5.1A,B Exercise #1 Classify as a monomial, binomial, or trinomial and give the degree. xy 3 z 4 – x 7 Binomial. Degree is determined by comparing Degree 1st term: x 1 y 3 z 4 1 + 3 + 4=8 Degree 2nd term: x 7 Degree 8 7 Chapter 5 Section 5.1D Exercise #5 Add 6x 3 + 8x 2 – 6x – 4 and 6 – 3x + x 2 – 3x 3 . METHOD 1 6x 3 + 8x 2 – 6x – 4 – 3x 3 + x 2 – 3x + 6 3x 3 + 9x 2 – 9x + 2 Add 6x 3 + 8x 2 – 6x – 4 and 6 – 3x + x 2 – 3x 3 . METHOD 2 6 x 3 + 8x 2 – 6 x – 4 + 6 – 3 x + x 2 – 3 x 3 = 6x 3 + 8x 2 – 6x – 4 + 6 – 3x + x 2 – 3x 3 = 6x 3 – 3x 3 + 8x 2 + x 2 – 6x – 3x – 4 + 6 = 3x 3 + 9x 2 – 9x + 2 Chapter 5 Section 5.1D Exercise #6 Subtract 8x 3 – 6x 2 + 5x – 3 from 5x 3 + 3x 2 + 3. METHOD 1 – 5x 3 + 3x 2 +3 8x 3 – 6x 2 + 5x – 3 Subtract 8x 3 – 6x 2 + 5x – 3 from 5x 3 + 3x 2 + 3. METHOD 1 – 5x 3 + 3x 2 +3 8x 3 – 6x 2 + 5x – 3 5x 3 + 3x 2 + +3 Subtract 8x 3 – 6x 2 + 5x – 3 from 5x 3 + 3x 2 + 3. METHOD 1 – 5x 3 + 3x 2 +3 8x 3 – 6x 2 + 5x – 3 + 5x 3 + 3x 2 +3 – 8x 3 + 6x 2 – 5x + 3 – 3x 3 + 9x 2 – 5x + 6 Subtract 8x 3 – 6x 2 + 5x – 3 from 5x 3 + 3x 2 + 3. METHOD 2 5 x 3 + 3x 2 + 3 – 8 x 3 – 6 x 2 + 5 x – 3 = 5x 3 + 3x 2 + 3 – 8x 3 + 6x 2 – 5x + 3 = 5x 3 – 8x 3 + 3x 2 + 6x 2 – 5x + 3 + 3 = – 3x 3 + 9x 2 – 5x + 6 Section 5.2 Multiplication of Polynomials OBJECTIVES A Multiply a monomial by a polynomial. OBJECTIVES B Multiply two polynomials. OBJECTIVES C Use the FOIL method to multiply two binomials. OBJECTIVES D Square a binomial sum or difference. OBJECTIVES E Find the product of the sum and the difference of two terms. OBJECTIVES F Use the ideas discussed to solve applications. RULES Multiplication of Polynomials If P, Q, and R are polynomials, P P Q =Q P Q R = P Q R USING FOIL To Multiply Two Binomials (x + a)(x + b) First Outside Inside Last 2 = x + bx + ax + ab 2 = x + (b + a)x + ab RULE To Square a Binomial Sum (x + a) 2 = (x + a)(x + a) 2 = x + 2ax + a 2 RULE To Square a Binomial Difference 2 (x – a) = (x – a)(x – a) = x 2 – 2ax + a 2 PROCEDURE Sum and Difference of Same Two Monomials 2 2 2 2 (x – a)(x + a) = x – a (x + a)(x – a) = x – a Chapter 5 Section 5.2B,C Exercise #8a Multiply. x – 2 x 2 – 4x – 5 METHOD 1 x 2 – 4x – 5 x –2 – 2x 2 + 8x + 10 x 3 – 4x 2 – 5x x 3 – 6x 2 + 3x + 10 Multiply. x – 2 x 2 – 4x – 5 METHOD 2 = x x 2 – 4x – 5 + – 2 x 2 – 4x – 5 = x 3 – 4x 2 – 5x – 2x 2 + 8x + 10 = x 3 – 4x 2 – 2x 2 – 5x + 8x + 10 = x 3 – 6x 2 + 3x + 10 Chapter 5 Section 5.2D Exercise #9b Multiply. 3x – 4y 2 x – a 2 = 3x 2 – 2(3x)(4y) + 4y 2 = 9x 2 – 24xy + 16y 2 = x 2 – 2ax + a 2 Chapter 5 Section 5.2E Exercise #10 Multiply. 3x + 4y 3x – 4y Product of Sum and Difference of Same Two Monomials x + a x – a = x 2 – a 2 2 = 3x – 4 y = 9x 2 – 16y 2 2 Section 5.3 The Greatest Common Factor and Factoring by Grouping OBJECTIVES A Factor out the greatest common factor in a polynomial. OBJECTIVES B Factor a polynomial with four terms by grouping. GREATEST COMMON FACTOR n ax is the Greatest Common monomial Factor (GCF) of a polynomial in x if: 1. a is the greatest integer that divides each coefficient. GREATEST COMMON FACTOR n ax is the Greatest Common monomial Factor (GCF) of a polynomial in x if: 2. n is the smallest exponent of x in all the terms. PROCEDURE Factoring by Grouping 1. Group terms with common factors using the associative property. PROCEDURE Factoring by Grouping 2. Factor each resulting binomial. PROCEDURE Factoring by Grouping 3. Factor out the binomial using the GCF, by the distributive property. Chapter 5 Section 5.3B Exercise #12 Factor completely. 6x 7 + 6x 5 + 15x 4 + 15x 2 = 3x 2 2x 5 + 2x 3 + 5x 2 + 5 = 3x 2 2x 3 x 2 + 1 + 5 x 2 + 1 = 3x 2 x 2 + 1 2x 3 + 5 = 3x 2 x 2 + 1 2x 3 + 5 Section 5.4 Factoring Trinomials OBJECTIVES A Factor a trinomial of the 2 form x + bx + c. OBJECTIVES B Factor a trinomial of the 2 form ax + bx + c using trial and error. OBJECTIVES C Factor a trinomial of the 2 form ax + bx + c using the ac test. PROCEDURE Factoring Trinomials 2 x + (b + a)x + ab = (x + a)(x + b) RULE The ac Test 2 ax + bx + c is factorable only if there are two integers whose product is ac and sum is b. Chapter 5 Section 5.4A,B,C Exercise #13b Factor completely. 2x 2 + xy – 10y 2 The ac Method Find factors of ac (–20) whose sum is (1) and replace the middle term (xy). = 2x 2 – 4xy + 5xy – 10y 2 = (2x 2 – 4xy) + (5xy – 10y 2) = 2x(x – 2y) + 5y(x – 2y) = 2x + 5y x – 2y Section 5.5 Special Factoring OBJECTIVES A Factor a perfect square trinomial. OBJECTIVES B Factor the difference of two squares. OBJECTIVES C Factor the sum or difference of two cubes. PROCEDURE Factoring Perfect Square Trinomials 2 2 2 x + 2 ax + a = (x + a) 2 2 2 x – 2 ax + a = (x – a) PROCEDURE Factoring the Difference of Two Squares 2 2 x – a = (x + a)(x – a) PROCEDURE Factoring the Sum and Difference of Two Cubes 3 3 2 2 x + a = (x + a)(x – ax + a ) 3 3 2 2 x – a = (x – a)(x + ax + a ) Chapter 5 Section 5.5A Exercise #15a Factor completely. 16x 2 – 24xy + 9y 2 2 2 2 x – 2ax + a = x – a Perfect Square Trinomial = 16x 2 – 2 12xy + 9y 2 = 4x – 3y 2 Chapter 5 Section 5.5 Exercise #16 Factor completely. 2 2 4 4 2 2 x – 16y = x – 4y Difference of Two Squares x 2 – a 2 = x + a x – a = x 2 + 4y 2 x 2 – 4y 2 Factor x 2 – 4y 2 = x 2 + 4y 2 x + 2y x – 2y Chapter 5 Section 5.5B Exercise #17 Factor completely. x 2 – 10x + 25 – y 2 2 2 x – 2ax + a = x – a 2 Perfect Square Trinomial 2 = x – 5 – y 2 Difference of Two Squares x 2 – a 2 = x + a x – a = x – 5 + y x – 5 – y = x + y – 5 x – y – 5 Chapter 5 Section 5.5c Exercise #18a Factor completely. 27x 3 + 8y 3 = 3x 3 + 2y 3 Sum of Two Cubes x 3 + a 3 = x + a x 2 – ax + a 2 = 3x + 2y 9x 2 – 6xy + 4y 2 Section 5.6 General Methods of Factoring OBJECTIVES A Factor a polynomial using the procedure given in the text. PROCEDURE A General Factoring Strategy 1. Factor out the GCF, if there is one. 2. Look at the number of terms in the given polynomial. PROCEDURE A General Factoring Strategy If there are two terms, look for: Difference of Two Squares 2 2 x – a = (x + a)(x – a) PROCEDURE A General Factoring Strategy If there are two terms, look for: Difference of Two Cubes 3 3 2 2 x – a = (x – a)(x + ax + a ) PROCEDURE A General Factoring Strategy If there are two terms, look for: Sum of Two Cubes 3 3 2 2 x + a = (x + a)(x – ax + a ) PROCEDURE A General Factoring Strategy If there are two terms, look for: The sum of two squares, 2 2 x + a is not factorable. PROCEDURE A General Factoring Strategy If there are three terms, look for: Perfect square trinomial 2 2 2 x + 2ab + b = (x + a) 2 2 2 x – 2ab + b = (x – a) PROCEDURE A General Factoring Strategy If there are three terms, look for: Trinomials of the form 2 ax + bx + c (a > 0) PROCEDURE A General Factoring Strategy Use the ac method or trial and error. If a < 0, factor out – 1 first. PROCEDURE A General Factoring Strategy If there are four terms: Factor by grouping. PROCEDURE A General Factoring Strategy 3. Check the result by multiplying the factors. Chapter 5 Section 5.6A Exercise #20b Factor completely. 48x 2y 2 – 72xy 3 + 27y 4 = 3y 2 16x 2 – 24xy + 9y 2 Perfect Square Trinomial x 2 – 2ax + a 2 = x – a 2 = 3y 2 4x 2 – 2 12xy + 3y 2 2 2 = 3y 4x – 3y Chapter 5 Section 5.6A Exercise #21 The ac Method Find factors of ac (–12) whose sum is (–11) and replace the middle term (–11xy). Factor completely: 9x 3y – 33x 2y 2 – 12xy 3 = 3xy [3x 2 – 11xy – 4y 2] = 3xy [3x 2 – 12xy + xy – 4y 2] = 3xy [ 3x 2 – 12xy + xy – 4y 2 ] = 3xy [ 3x x – 4y + y x – 4y ] = 3xy [ x – 4y 3x + y ] The ac Method Find factors of ac (–12) whose sum is (–11) and replace the middle term (–11xy). Factor completely: 9x 3y – 33x 2y 2 – 12xy 3 = 3xy [3x 2 – 11xy – 4y 2] = 3xy [3x 2 – 12xy + xy – 4y 2] = 3xy [ 3x 2 – 12xy + xy – 4y 2 ] = 3xy [ 3x x – 4y + y x – 4y ] = 3xy 3x + y x – 4y Chapter 5 Section 5.6A Exercise #22 Factor completely. 16x 3 – 12x 2 – 4x + 3 = 4x 2 4x – 3 – 1 4x – 3 = 4x – 3 4x 2 – 1 Difference of Two Squares x 2 – a 2 = x + a x – a = 4x – 3 2x + 1 2x – 1 Section 5.7 Solving Equations by Factoring: Applications OBJECTIVES A Solve equations by factoring. OBJECTIVES B Use Pythagorean theorem to find the length of one side of a right triangle when the lengths of the other two sides are given. OBJECTIVES C Solve applications involving quadratic equations. PROCEDURE 1. Set equation equal to 0. O 2. Factor Completely. F 3. Set each linear Factor F equal to 0 and solve each. DEFINITION Pythagorean Theorem In symbols, 2 2 2 c =a +b a c b Chapter 5 Section 5.7A Exercise #23b Solve. O 6x 2 + 7x = 3 6x 2 + 7x – 3 = 0 F 3x – 1 2x + 3 = 0 F 3x – 1 = 0 3x = 1 1 x= 3 or or 2x + 3 = 0 2x = – 3 3 x=– 2 1 3 x= , – 3 2