OCF.01.2 - Operations With Polynomials

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OCF.01.2 - Operations With Polynomials. MCR3U - Santowski. (A) Review. Like terms are terms which have the same variables and degrees of variables ex: 2x, -3x, and 2 x are like terms while 2xy is not a like term with the other x terms

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### OCF.01.2 - Operations With Polynomials

MCR3U - Santowski

(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: 2x2, -3x2, and 2x2 are like terms while 2x3 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
• 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 (4x2 - 7x - 5) + (2x2 - x + 3)
• Simplify (4s2 + 5st - 7t2) - (6s2 + 3st - 2t2)
(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(a3 - 4a - 5)
• (ii) Expand and simplify 2x(3x - 5) - 4x(x - 7) + 3x(x - 1)
(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 (x2 - 3x - 1)(2x2 + x - 2)
(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
• (ii) factor by grouping  pair the terms that have a common factor
• (iii) simple inspection  useful for simple trinomials
• (iv) decomposition  useful for more complex trinomials
(E) Examples of Factoring
• Ex. Factor 60x2y – 45x2y2 + 15xy2 identify the GCF of 15xy
• 15xy(4x – 3xy + y)
• Ex Factor 3xy – 5xy2 + 6x2y – 10x2y2
• 3xy + 6x2y – (5xy2 + 10x2y2)
• 3xy(1 + 2x) – 5xy2(1 + 2x)
• (3xy – 5xy2)(1 + 2x)
• xy(3 – y)(1 + 2x)
(E) Examples of Factoring
• Ex. Factor x2 – 3x – 10
• 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
• Therefore x2 – 3x – 10 = (x - 5)(x + 2)
(E) Examples of Factoring
• Ex. Factor 4x2 – 14x - 8 using the decomposition technique
• First multiply 4x2 by 8 to get -32x2
• Now consider the middle term of -14x
• Now find two terms whose product is -32x2 and whose sum is -14x  the terms are -16x and 2x
• Replace the -14x with -16x + 2x
• Then we have the “decomposed” expression 4x2 – 16x + 2x – 8 and now we simply factor by grouping
• 4x(x – 4) + 2(x – 4)
• (4x + 2)(x – 4)
• 2(2x + 1)(x -4)
(D) Homework
• AW, pg87-89
• Q8,15,20,21,23,24
• Nelson Text  p302, Q2,3,5,7