module 3 math dba notes

03.01LessonSummary
To achieve mastery of this lesson, make sure that you develop responses to the essential question listed below.
How can a Greatest Common Factor be separated from an expression?
By simplifying the equation . By breaking them up by dividing them up
What methods can be used to rewrite square trinomials and difference of squares binomials as separate factors? distribution in what conditions can a factored expression be factored further?
Greatest Common Factor
A greatest common factor of two or more terms is the largest factor that all terms have in common. The greatest common factor of a polynomial should be factored out first before any further factoring is completed.
Example:
3r6+27r4+15r2=3r2(r4+9r2+5)

When multiplying variables, add the exponents. r^2•r^4=rr•rrrr=r6 When factoring a GCF, subtract the exponents.
To factor r^2 from r^6: r^6−2=r^4 rrrrrr=(rr)(rrrr)=r2(r4) Difference of Squares Binomials
A difference of squares binomial includes a perfect square term subtracted by another perfect square term.
Pattern:
a^2−b^2=(a+b)(a−b)
Example:
r^2−4=(r+2)(r−2)
Perfect Square Trinomials
A perfect square trinomial is a polynomial of three terms where the first and last terms are perfect squares and the middle term is twice the product of the square roots of those terms.

Pattern: a^2+2ab+b^2=(a+b)^2 OR a^2−2ab+b^2=(a−b)^2
Example:
r^2+12r+36=(r+6)^2 r^2−12r+36=(r−6)^2

Check your factors to see if they can be factored further
Sometimes after an initial factoring, the remaining terms can be factored further.
Example:
t^8−81
This is a difference of squares.
(t^4+9)(t^4−9)
The factor(t^4−9) is also a difference of squares.
(t^4+9)(t2+3)(t^2−3)
Even though there are even powers here, these are not special products so this is factored completely.
Example:
3w^2+15w+18
This has a GCF of 3.
3(w^2+5w+6)
The trinomial can be factored by looking for factors of 6 that add up to 5.
3(w+3)(w+2)
This is completely factored.

03.02LessonSummary
To achieve mastery of this lesson, make sure that you develop responses to the following essential questions:
How can a Greatest Common Factor be separated from an expression?

What methods can be used to rewrite square trinomials and difference of squares binomials as separate factors?

In what conditions can a factored expression be factored further?
Discriminant Information
This is the standard form of a trinomial:ax2+bx +c
The Discriminant is:b2− 4ac
The nature of the roots can be determined by substituting the numbers from the trinomial into the Discriminant and solving.
If the Discriminant is:
Then there will be:
Zero
1 rational solution
Positive Perfect Square
2 rational solutions
Positive Integer, Not a Perfect Square
2 irrational solutions
Negative
2 complex solutions

Steps for Factoring Polynomials with Four Terms
1. Factor the GCF.
Determine if there is a GCF among all terms in the original polynomial and factor it, if one exists.
2. Group the terms.
Group the polynomial into two pairs—the first two terms and the last two terms.
3. Factor the GCF from each group.
Factor the GCF from the first pair of terms and then from the second pair of terms.
4. Factor the common binomial.
If common binomials do not exist, GCFs were factored accurately, and the expression cannot be rearranged, you have a prime polynomial.
Don’t forget to check the factors for accuracy by distributing.

Steps for Factoring Trinomials by Grouping
1. Factor the GCF.
Determine if there is a GCF among all terms in the original polynomial and factor it, if one exists.
2. Split the middle term.
Multiply the leading coefficient and the constant.
Find factors of this product that sum to the middle coefficient.
Rewrite the polynomial with those factors replacing the middle term.
3. Factor by grouping
The four-term polynomial is split into two groups—the first pair and last pair of terms. Factor the GCF from each pair.
Factor the common binomial.
If common binomials do not exist and GCFs were factored accurately, you may have a prime polynomial.
4. Check your work.
Check the factors by distributing.

03.03LessonSummary
To achieve mastery of this lesson, make sure that you develop responses to the essential question listed below:
Using the structure of the expression given, how can sums and differences of cubes be factored?
Sum of Cubes Pattern:a3+ b3= (a + b)(a2− ab + b2)
Difference of Cubes Pattern:a3− b3= (a − b)(a2+ ab + b2)

Steps to Factoring the Sum/Difference of Cubes
1. Factor the GCF.
2. Identify the cube root of each term.
The cube root of the first term will represent a in the pattern.
The cube root of the second term will represent b in the pattern.
3. Substitute a and b in the appropriate pattern.
Don’t forget to check the factors by multiplying!

03.04LessonSummary
A quadratic equation is an equation of degree 2. The graph of a quadratic equation is in the shape of a parabola which looks like an arc. The general form of the equation is represented by f(x) = a(x − h)2+ k
The vertex, or turning point, of the parabola is an ordered pair represented by (h, k). The vertex of the quadratic equation f(x) = −(x − 2)2 + 9 is (2, 9).
Using the average rate of change, it is possible to find the slope between two points and observe if the graph’s rate of change is increasing or decreasing.
The axis of symmetry is a line which splits the parabola in half so that both halves are symmetrical to one another. The equation for this line is found by setting the expression inside the parentheses equal to 0 and solving for x. f(x) = −(x − 2)2+ 9 x − 2 = 0
+2 +2 x = 2
The domain of the quadratic equation is the set of numbers which can be substituted for x and result in a unique value for y. In the case of the equation shown here, the domain is "all real numbers".
The range of the quadratic equation is the set of numbers that are produced from the domain values of x. Algebraically, determine whether the vertex is the minimum or maximum point of the graph. If a is positive, the vertex is the minimum point, and the range is all of the y-coordinates greater than or equal to the y-coordinate of the vertex. If a is negative, the vertex is the maximum point, and the range is all of the y-coordinates less than or equal to the y-coordinate of the vertex.

Graphically, it is easy to see the vertex is a maximum point and y-coordinates greater than 9 do not have a corresponding x-coordinate. Therefore, in the case of the equation shown, the range is y ≤ 9.
The intercepts of a quadratic equation are the places on the graph where the parabola crosses the x- or y-axis. The x-intercepts are found by looking at the graph of the parabola where it crosses the x-axis.
The standard form of the quadratic equation is represented by f(x) = ax2 + bx + c

While the domain, range and intercepts of the parabola may be found in the same way as the general form of the parabola, the axis of symmetry and vertex must be found using the formula

The axis of symmetry is found first using this equation. f(x) = x2+ 6x + 8

x =−3

Then the vertex is found by substituting the x-coordinate of −3 in the original equation and solving for y to find the y-coordinate. f(x) = x2+ 6x + 8 f(x) =(−3)2+ 6(−3)+ 8 f(x) =9+ 6(−3) + 8 f(x) =9 − 18+ 8

f(x) = −1(−3, −1)

03.06LessonSummary
To achieve mastery of this lesson, make sure that you develop responses to the essential questions listed below.
1. How can completing the square aid in solving quadratic functions? It helps you find the maximum/minimum at a certain x-value. Thus turning point. Because you can transform a quadratic form ( with x^2) into 2 binomial forms (ax + b) that you can easily solve for x.

1. What indicators predict that a quadratic function will have a complex solution?
When the Discriminant D = b^2 - 4ac is negative. Reminder; D = b^2 - 4ac > 0: there are 2 real roots D = 0: double root at x = -b/2a D < 0: no real roots, there are complex roots. Reminder. When D is a perfect square, the quadratic equation can be factored.
Steps for Changing a Quadratic Equation from Standard Form to General Form
1. Isolate the terms containing the x variable.
2. Complete the square and balance the equation.
Factor the numerical GCF to ensure the leading coefficient of the binomial is 1.
Divide the coefficient of x by 2 and square the result to create a perfect square trinomial.
To keep the equation balanced, the constant added to one side of the equation must also be added to the other side of the equation.

3. Simplify.
Combine like terms.
Factor the perfect square trinomial.
Isolate the y variable.

4. Check your work.
Expand and multiply the squared binomial and then combine like terms.
Check the Discriminant to find the nature of the solutions
The Discriminant is b2 − 4ac.
Discriminant
If the Discriminant is:
Then there will be:
Zero
1 rational solution
Positive Perfect Square
2 rational solutions
Positive Integer, Not a Perfect Square
2 irrational solutions
Negative
2 complex solutions
Steps for Solving a Quadratic Equation by Completing the Square
1. Isolate the terms containing the x variable.
2. Complete the square and balance the equation.
Divide the coefficient on x by 2 and square the result to create a perfect square trinomial.
To keep the equation balanced, the constant added to one side of the equation must also be added to the other side of the equation.

3. Simplify.
Combine like terms.
Factor the perfect square trinomial.

4. Solve for the variable.

03.07 Lesson Summary
To achieve mastery of this lesson, make sure that you develop responses to the essential questions listed below.
What methods can be used to solve quadratic equations?
How are solutions, roots, and x-intercepts of a quadratic related?
Steps for Solving a Quadratic Equation by Factoring
1. Write the equation in standard form.
2. Factor the quadratic expression.
3. Set each factor equal to 0 and solve.
4. Check all solutions.
Steps for Solving a Quadratic Equation using the Quadratic Formula
3x2− 12x = 27
1. Rewrite the equation in standard form.
3x2− 12x = 27
−27 −27
3x2− 12x − 27 = 0

2. Factor any existing GCF.
3(x2 − 4x − 9) = 0

3. Apply and simplify the quadratic formula.
3(x2 − 4x − 9) = 0

a = 1, b = −4 and c = −9

4. Check your work. x = x =
3x2 − 12x = 27

27= 27
3x2 − 12x = 27

27= 27

03.08 Lesson Summary
To achieve mastery of this lesson, make sure that you develop responses to the essential question listed below.
When the solutions are going to be complex, which methods can be used to solve quadratic equations?

Steps for Solving a Quadratic Equation with Complex Numbers
Completing the Square
1. Isolate the terms containing the x variable.
2. Complete the square and balance the equation.
Factor the numerical GCF to ensure the leading coefficient on x2 is 1.
Divide the coefficient on x by 2 and square the result to create a perfect square trinomial.
To keep the equation balanced, the constant added to one side of the equation must also be added to the other side of the equation.
3. Simplify.
Factor the perfect square trinomial.
Combine like terms.
4. Solve for the variable.
Quadratic Formula
1. Write the equation in standard form.
2. Factor the GCF.
3. Apply and simplify the quadratic formula.

03.09LessonSummary
To achieve mastery of this lesson, make sure that you develop responses to the essential question listed below.
1. How can the equation of a parabola be derived when given the focus and the directrix?

Steps for Deriving a Quadratic Equation from a Parabola Using Distances.
Step 1
Substitute information into combined distance formulas. The right side of the equation is what was derived for the directrix.

Step 2
Remove radicals.

Step 3
Distribute y term binomials.

Step 4
Simplify and isolate x terms.

Step 5
Isolate the y term.
Steps for Deriving a Quadratic Equation from a Parabola Using Focal Length.

Step 1
Sketch a graph to locate vertex and focal lengths.

Step 2
Solve for the a coefficient. = a

Step 3
Substitute vertex and a coefficient into vertex form.