OCF.01.2 - Operations With Polynomials

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OCF.01.2 - Operations With Polynomials

• MCR3U - Santowski

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(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
•  

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(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 (4x2 - 7x - 5) (2x2 - x 3)
• Simplify (4s2 5st - 7t2) - (6s2 3st - 2t2)
•  

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(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)

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(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)

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(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

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(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)

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(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)

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(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)

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(D) Homework

• AW, pg87-89
• Q8,15,20,21,23,24
• Nelson Text ? p302, Q2,3,5,7