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 - 6n 3 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  3x 2 – 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

4. (C) Examples ex. Factor 6x 3 + 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

6. (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. 3x 2 - 21 = 2x ex. 5a 2 + 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 +  (b 2 – 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

7. (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

8. (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

9. (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)

10. (I) Internet Links College Algebra Tutorial on Factoring Polynomials from West Texas A&M College Algebra Tutorial on Factoring Polynomials from West Texas A&M College Algebra Tutorial on Quadratic Equations 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 from OJK's Precalculus Page Solving Quadratic Equations Lesson - from Purple Math Solving Quadratic Equations Lesson - from Purple Math

11. (I) Homework Nelson Text, p4, Q1-14