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1.1 – Reviewing Functions Algebra
MCB4U - Santowski
(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
(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
(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)
(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
(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
(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
(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
(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)
(I) Internet Links
College Algebra Tutorial on Factoring
Polynomials from West Texas A&M
College Algebra Tutorial on Quadratic Equations
from West Texas A&M
Solving quadratic equations from OJK's
Precalculus Page
Solving Quadratic Equations Lesson - from
Purple Math
(I) Homework
Nelson Text, p4, Q1-14