1 OCF Operations With Polynomials MCR3U - Santowski

2 (A) Review Like terms are terms which have the same variables and degrees of variables ex: 2x, -3x, and 2x are like terms while 2xy is not a like term with the other x terms ex: 2x 2, -3x 2, and 2x 2 are like terms while 2x 3 is not a like term with the other squared terms A monomial is a polynomial with one term A binomial is a polynomial with two terms A trinomial is a polynomial with three terms

3 (B) Adding and Subtracting Polynomials Rule: Addition and subtraction of polynomials can only be done if the terms are alike in which case, you add or subtract the coefficients of the terms Simplify (4x 2 - 7x - 5) + (2x 2 - x + 3) Simplify (4s 2 + 5st - 7t 2 ) - (6s 2 + 3st - 2t 2 )

4 (C) Multiplying Polynomials When multiplying a polynomial with a monomial (one term), use the distributive property to multiply each term of the polynomial with the monomial Exponent Laws : When multiplying two powers, you add the exponents (i) Expand 3a(a 3 - 4a - 5) (ii) Expand and simplify 2x(3x - 5) - 4x(x - 7) + 3x(x - 1)

5 (C) Multiplying Polynomials When multiplying a polynomial with a binomial, you can also use the distributive property. (i) Expand and simplify (2x + 3)(4x - 5) Show two ways of doing it (ii) Expand and simply (x + 4) 2 (iii) Expand and simplify (3x + 5)(3x - 5) (iv) Expand and simplify 3x(x - 4)(x + 2) - 2x(x + 5)(x - 3) (v) Expand and simplify (x 2 - 3x - 1)(2x 2 + x - 2)

6 (D) Factoring Algebraic Expressions Polynomials can be factored in several ways: (i) common factoring  identify a factor common to the various terms of the algebraic expression (i) common factoring  identify a factor common to the various terms of the algebraic expression (ii) factor by grouping  pair the terms that have a common factor (ii) factor by grouping  pair the terms that have a common factor (iii) simple inspection  useful for simple trinomials (iii) simple inspection  useful for simple trinomials (iv) decomposition  useful for more complex trinomials (iv) decomposition  useful for more complex trinomials

7 (E) Examples of Factoring Ex. Factor 60x 2 y – 45x 2 y xy 2  identify the GCF of 15xy 15xy(4x – 3xy + y) 15xy(4x – 3xy + y) Ex Factor 3xy – 5xy 2 + 6x 2 y – 10x 2 y 2 Ex Factor 3xy – 5xy 2 + 6x 2 y – 10x 2 y 2 3xy + 6x 2 y – (5xy x 2 y 2 ) 3xy + 6x 2 y – (5xy x 2 y 2 ) 3xy(1 + 2x) – 5xy 2 (1 + 2x) 3xy(1 + 2x) – 5xy 2 (1 + 2x) (3xy – 5xy2)(1 + 2x) (3xy – 5xy2)(1 + 2x) xy(3 – y)(1 + 2x) xy(3 – y)(1 + 2x)

8 (E) Examples of Factoring Ex. Factor x 2 – 3x – 10 The trinomial came from the multiplication of two binomials  since the leading coefficient is 1, therefore (x )(x ) is the first step The trinomial came from the multiplication of two binomials  since the leading coefficient is 1, therefore (x )(x ) is the first step Then, find which two numbers multiply to give -10 and add to give -3  the numbers are -5 and +2 Then, find which two numbers multiply to give -10 and add to give -3  the numbers are -5 and +2 Therefore x 2 – 3x – 10 = (x - 5)(x + 2) Therefore x 2 – 3x – 10 = (x - 5)(x + 2)

9 (E) Examples of Factoring Ex. Factor 4x 2 – 14x - 8 using the decomposition technique First multiply 4x 2 by 8 to get -32x 2 First multiply 4x 2 by 8 to get -32x 2 Now consider the middle term of -14x Now consider the middle term of -14x Now find two terms whose product is -32x 2 and whose sum is - 14x  the terms are -16x and 2x Now find two terms whose product is -32x 2 and whose sum is - 14x  the terms are -16x and 2x Replace the -14x with -16x + 2x Replace the -14x with -16x + 2x Then we have the “decomposed” expression 4x 2 – 16x + 2x – 8 and now we simply factor by grouping Then we have the “decomposed” expression 4x 2 – 16x + 2x – 8 and now we simply factor by grouping 4x(x – 4) + 2(x – 4) 4x(x – 4) + 2(x – 4) (4x + 2)(x – 4) (4x + 2)(x – 4) 2(2x + 1)(x -4) 2(2x + 1)(x -4)

10 (D) Homework AW, pg87-89 Q8,15,20,21,23,24 Nelson Text  p302, Q2,3,5,7