Basic Factoring
1) List the three types of factoring (may use examples):
2) Which type of factoring should you always look for first?
3) 4ab2 –12a2b 4) y2 – 12y + 36 5) 16x2 – 25
6) 36c2 – 100d2 7) x2 + 2x – 24 8) x2 – x – 6
9) 10)
11) What is special about numbers 9 and 10?
12) How would you simplify the following? 7x(x + 2) – 8(x + 2)
13) 14)
* the process used in 13 & 14 is called factoring by grouping
The AC Method
Warm Up:ADD/MULTIPLY
1) Find two numbers whose sum is –7 and product is –60?
2) Find two numbers whose sum is –21 and product is –72?
Step by Step method for factoring Ax2 + Bx + C
Step 1: Multiply together AC and list the factors of AC.
Step 2: Find a pair that adds to B. If you cannot find such a pair then the trinomial does not factor.
Step 3: Rewrite the middle term as a sum of terms whose coefficients are the chosen pair.
Step 4: Factor by grouping.
Remember you should always first pull out the GCF
Example:
Factor 15x2 - x - 6
AC = (15)(-6) = -90
List all the factors of -90: (1, -90), (-1, 90),
We see thatB = -1 so the correct pair of factors must be:
Now Rewrite the original polynomial: 15x2 - x – 6
Replace the x term with the pair from above:
Factor first two terms by GCF then second two terms:
Factor by grouping:
Factor the following using the ac method:
1) 2x2 – x – 6 2) 6x 2 + 5x – 6
3) 6x2 + 7x + 2 4) 4x4 + 8x2 + 3
Solving
1) What are the differences between solving and factoring?
2) What are some methods for solving a quadratic?
3) Solve the following equation by factoring: 7x2= 20x – 12
4) Solve the following equation by factoring: 30x3 + 25x2 + 5x= 0
More Solving
Quadratic Formula:
Solve the following using the quadratic formula and express your answer in simplest radical form.
1)
2)
3)
For each of the above:
§ Describe the roots (real, rational, imaginary, equal)
§ State how many time the parabola will intersect the x – axis
Completing the Square
Warm Up
1) Factor:
2) What kind of trinomial is in number 1?
/ StepsTransform the equation so that only terms with variables are on the left hand side. If a > 1 then divide through by a.
Add to both sides in order to form a perfect square trinomial on the left.
Show the square of the binomial on the left; simplify the number on the right.
Take the square root of both sides, remember to put a on the right.
Solve for x to find the roots
Solutions are a ______pair.
1)
Solve each quadratic equation by completing the square; express each root in simplest radical form.
2) 3)
4) 5)
6)
For each of the above:
§ Describe the roots (real, rational, imaginary, equal)
§ State how many time the parabola will intersect the x – axis
7) Solve the quadratic equation by completing the square:
Quad Apps
1) A model rocket is launched from ground level. Its height, h meters above the ground, is a function of time t seconds after launch and is given by the equation h = -4.9t2 + 68.6t. At what time will the rocket hit the ground? (solve using factoring)
2) A ball is thrown straight up at an initial velocity of 54 feet per second. The height of the ball t seconds after it is thrown is given by the formula h(t) = 54t –12t2. How many seconds after the ball is thrown will it return to the ground? (solve using CTS)
3) An archer shoots an arrow into the air such that its height, in feet, at any time, t, is given by the function h(t) = -16t2 – 128t + 3. At what time will the arrow hit the ground? (solve using quad form)
Rational Expressions
Simplify:
1) 2)
Multiply
§ Example
§ Factor everything
§ Cancel top and bottom
§ Multiply Across =
3)
Divide
§ Same as multiplication except keep change flip first.
4)
Add/Subtract
§ Example
§ Factor denominators
§ Find common denominator *can’t cancel here
§ Add/subtract numerators
§ Simplify =
§ Be careful when combining fractions under subtraction! Be sure to subtract the ENTIRE numerator value behind the subtraction sign. In this problem, when “subtracting”
(x – 6) it is necessary to distribute the negative (subtraction) sign across the parentheses, creating –x + 6.
5) 6)
Undefined Fractions
1) What makes a fraction undefined?
2) For what value(s) of x are the following undefined?
a. b.
c.
Fractional Equations
3) Solve for x: List Steps:
1.
2.
3.
4.
5.
4)
What are the differences between a rational expression and a rational (fractional) equation?
Complex Fractions
5) Steps1.
2.
3.
4.
6) Your turn:
1