1.1

1
1.1 Reviewing Functions - Algebra

• MCB4U - Santowski

2
(A) Review of Factoring Common Factoring

• NOTE Write answers in 2 forms once as a product
  and once as a sum/difference.
• - factor 6ap - 24aq (common factor is ..? )
• - factor - 5axy - 5bxy 10cxy
• - factor 3tX 7X
• - factor 3t(a b) 7(a b)
• - factor 2a(m - n) (-m n)
• - factor 2ax - bx 6ay - 3by
• - factor 10x2 3y - 5xy - 6x

3
(B) Factoring Trinomials - aka Quadratic
Expressions

• - a quadratic expression is a polynomial of
  degree 2 in the form of ax² bx c
• 1. Factor by inspection (usually when a 1)
• ex. Factor y² 9y 14 ? what multiplies to 14
  and adds to 9?
• ex. Factor m²n - mn² - 6n3
• 2. Factor by decomposition (usually if a is not
  equal to 1)
• ex. Factor 3x² - 7x - 6 ? the middle term of -7x
  is decomposed into -9x 2x ? 3x2 9x 2x 6
  ? then factor by grouping
• point out the guess and check method - consider
  the factors of 3 and consider the factors of -6
  and try to find the combination that gives you a
  -7x as the middle term

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(C) Examples

• ex. Factor 6x3 x² - 2x
• ex. Factor 6x² - 11x - 10
• ex. Factor 9m² 33m 30
• ex. Factor 8t² 4t 4
• Recall the graphical interpretation of the
  solution ? graph the expressions as equations on
  WINPLOT or a GDC ?(roots, zeroes, x-intercepts)

5
(D) Factoring Perfect Square Trinomials and
Difference of Squares

• (i) Factoring Perfect Square Trinomials
• use decomposition to see the pattern, then simply
  use the pattern in the future
• factor 25m² 40nm 16n²
• factor 36s² 120s 100
• (ii) Factoring Difference of Squares
• use decomposition to see the pattern (middle term
  is 0x), then simply use the pattern in the
  future
• factor 4x² - 9
• factor 18d² - 50f²
• factor (x - y)² - 16
• - factor by grouping to show a difference of
  squares x² 6xy 9y² - 36
• - factor -x² y² 6yz 9z² 4x 4

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(E) Review of Solving Quadratic Equations

• quadratic equations are equations in the form of
  0 ax² bx c
• some quadratic equations can be factored over the
  integers in which case we can solve by factoring
• ex. 3x2 - 21 2x
• ex. 5a2 45 -30a
• Now use WINPLOT or a GDC to visualize the
  solution
• some QE cannot be factored so there must be
  another method of solving these equations
• ex. 0 2x² 5x 1
• so we will use the quadratic formula which is
  -b ?(b2 4ac) ?2a
• We can also use the completing the square
  method to isolate the variable
• Additionally, we can simply using graphing
  technology to graph y ax² bx c and find the
  zeroes, roots, x-intercepts

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(F) Examples

• Solve and graph 3x² - 21 2x. Find the roots of
  3x² - 21 2x
• Solve g(a) 5a² 45 30a. Graph and find the
  roots of g(a)
• Solve 3x² - 4x 7 13. Graph and find the
  x-intercepts of 3x² - 4x 7 13

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(G) Review of Complex Numbers

• Solve the equation x² 1 0.
• Regardless of the method we chose to employ, we
  come up with the problem that we cannot find a
  real number that satisfies the equation x² -1.
• to resolve this problem, mathematicians have
  developed another number system that will take
  into account the idea of a square root of a
  negative number.
• so we introduce a symbol, called the imaginary
  unit, i, which has the property that i² -1 or i
  ?(-1)
• so to solve a problem like x² 4 0
• x² -4
• x² (-1)(4)
• x 2i

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(H) Examples

• Simplify ?(-121)
• Simplify ?(-50)
• Solve the quadratic equation x² 2x 5 0 (x
  -1 2i) use GC to se what graph looks like
• So complex numbers have the form a bi and its
  conjugate would be a - bi
• Simplify (3 - 2i) 3(2 6i)
• Simplify (4 - 3i)²
• Simplify (4 - 3i)(3 - 5i)

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(I) Internet Links

• College Algebra Tutorial on Factoring Polynomials
  from West Texas AM
• College Algebra Tutorial on Quadratic Equations
  from West Texas AM
• Solving quadratic equations from OJK's
  Precalculus Page
• Solving Quadratic Equations Lesson - from Purple
  Math

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(I) Homework

• Nelson Text, p4, Q1-14