Midterm Review- Algebra 2

Factoring GCF, Difference of Squares, and Trinomials Review

Factoring a GCF

Review

Steps to Factor Using the GCF

• Find the GCF of the numbers (use Desmos if you need).

• Find the GCF of the variables. If a variable is in EVERY term, use the one with the smallest exponent.

• Write the GCF in front of parentheses.

• Divide each term in the problem by the GCF (dividing the coefficients and subtracting the exponents), and write the remainder inside the parentheses.

Multiple Choice

Factor:
$$5a^2-15$$  

• $$5\left(a^2-15\right)$$  

• $$5a\left(a-15\right)$$  

• $$5\left(a^2-3\right)$$  

• $$5a\left(a-3\right)$$  

Multiple Choice

$$12c^2-20a^2$$  

Factor:

• $$2\left(6c^2-10a^2\right)$$  

• $$4\left(3c^2-5a^2\right)$$  

• $$2a^2c^2\left(6-10\right)$$  

• $$4\left(ca\right)^2\left(3-5\right)$$  

Multiple Choice

Factor the polynomial.

   $$50x^3+60x^2-60$$  

•     $$5\left(10x^3+12x^2-12\right)$$  

•     $$10\left(5x^3+6x^2-6\right)$$  

•     $$10x\left(5x^3+6x^2-6\right)$$  

•     $$10x\left(5x^2+6x-6\right)$$  

Multiple Choice

Factor the polynomial.

       $$77x^9-21x^7+49x^5$$  

•         $$7x^5\left(11x^4-3x^2+7\right)$$  

•         $$7x^5\left(11x^5-3x^2+7\right)$$  

•         $$7\left(11x^9-3x^7+7x^5\right)$$  

•         $$7x\left(11x^8-3x^6+7x^4\right)$$  

Factoring Difference of Squares

Review

Steps to Factor with Difference of Squares

• Check to make sure you have a difference of squares! You should have two perfect squares separated by a subtraction sign.

• Take the square root of each term.

• Write your two factors using the rule: a²-b² = (a+b)(a-b).

Multiple Choice

x² - 9

• (x + 3) (x - 3)
• (x - 3) (x - 3)
• (x + 3) (x - 6)
• Can't factor using Diff. of Squares

Multiple Choice

4a² - 25

• (2a + 5) (2a - 5)
• (2a - 5) (2a - 5)
• (x + 5) (x - 5)
• Can't factor using Diff. of Squares

Multiple Choice

4m² + 49

• (2m + 7) (2m - 7)
• (2m - 7) (2m - 7)
• (m + 7) (m - 7)
• Can't factor using Diff. of Squares

Factoring a Trinomial with a=1

Review

Steps to Factoring Trinomial (a=1)

• Trinomial: ax²+bx+c

  Goal: Find factors of c that add up to b.

• Use the graphic organizer flow chart to figure out the signs of the numbers.

• If needed, make a table of possible factors that multiply to your c value. Find the pair that add up to b.

Multiple Choice

Factor completely.
k² -2k - 24

• (k+4)(k-6)
• (k+4)(k+6)
• (k+6)(k-1)
• (k-4)(k+6)

Multiple Choice

Factor completely: x²-6x+8

• (x-2)(x-4)
• (x-1)(x-8)
• (x+2)(x+4)
• (x+1)(x+8)

Multiple Choice

Factor completely: x²-13x+40

• (x-5)(x-8)
• (x+5)(x+8)
• (x-4)(x-10)
• (x-2)(x-20)

Steps to Factor a GCF out First

• Factor the GCF following the same methods as before (find the GCF of the numbers, of the variables, the divide it from each term)

• If what is left in parentheses is a difference of squares, factor what is in parentheses using that method. Write the GCF in front.

• If what is left in parentheses is a trinomial, factor what is in parentheses using that method. Write the GCF in front.

GCF with Difference of Squares

1. Factor out GCF

2. Factor remaining difference of squares

GCF with Trinomial (a=1)

1. Factor out GCF

2. Factor remaining trinomial

Multiple Choice

$$125x^2-5$$  

Factor out the GCF first. Then, factor using another method.

• $$5\left(3x+4\right)\left(3x-4\right)$$  

• $$5\left(5x+1\right)\left(5x-1\right)$$  

• $$\left(5x-1\right)^2$$  

• $$5\left(25x+1\right)^2$$  

Multiple Choice

Factor completely.
3x² - 9x -120

• 3(x - 8)(x + 5)
• 3(x + 8)(x - 5)
• (3x + 8)(x - 5)
• (3x - 8)(x + 5)

Factoring when a>1

Watch this video for a review: https://www.youtube.com/watch?v=z-80Tlx0jEc

Multiple Choice

Factor:
 2m² + 3m - 9

• 2(m + 3)(m - 3)
• (2m - 3)²
• (m + 3)(2m - 3)
• (2m + 3)(m - 3)

Multiple Choice

Factor Completely:  6x² + 5x - 6

• 6(x + 1)(x - 6)
• (6x + 6)(x - 1)
• (2x - 3)(3x + 2)
• (3x - 2)(2x + 3)

Multiple Choice

Factor
3v² - 4v - 7

• (3v-7)(v+1)
• 3(v-7)(v-1)
• (3v+1)(v-9)
• (3v+1)(v-10)

Solving Quadratics by Square Roots

Review

To solve quadratic equations, we use inverse (opposite) operations

​Day 2

​Remember:

​Complex numbers are written in the form a + bi, where a and b are real numbers; a is the real part and bi is the imaginary part.

Complex Numbers

Learning Target: I can simplify expressions with complex numbers.

Adding Complex Numbers

• When adding complex numbers

  -Add real parts together

  -Add imaginary parts together

Multiple Choice

Find the Sum

3i + 2i

• 5i

• 5i²

• 6i

• -5

Subtracting Complex Numbers

• Distribute the negative

• Add real parts together

  Add imaginary parts together

Multiple Choice

Simplify:

3 + 2i + 4i + 6

• 9 + 6i

• 7i + 8

• 9 + 6i²

• 9 - 6i

Multiple Choice

Simplify:

4 + 7i - 3 - 2i

• 1 + 5i

• 7 + 9i

• 1 + 9i

• -1 + 5i

Complete the Square

Find the c value by completing the square

• b divided by 2

• then square it

The Quadratic Formula

a guided tour...

Goal #1:

I can identify a, b, and c when given a quadratic equation.

Match

What are the values of a, b, and c for the quadratic:

$$x^2+7x+8$$

• a =

• b =

• c =

1

7

8

• 1

• 7

• 8

Match

What are the values of a, b, and c for the quadratic:

$$11\ -\ 2x\ +17x^2$$

• a =

• b =

• c =

17

-2

11

• 17

• -2

• 11

Match

What are the values of a, b, and c for the quadratic:

$$-13x^2+8-6x$$

• a =

• b =

• c =

-13

-6

8

• -13

• -6

• 8

Goal #2:

I can substitute a, b, and c into the quadratic formula.

Labeling

Drag and drop the values of a, b, and c into the quadratic formula for:

$$x^2+14x-32$$

tip: values without ( ) go into the formula first, values with ( ) go in second.

Drag labels to their correct position on the image

1

(1)

(14)

-32

14

Goal #3:

I can simplify what is underneath the radical (called the discriminant).

Math Response

Simplify:

$$\sqrt[]{\left(-5\right)^2-4\left(1\right)\left(5\right)}$$

Type answer here

Correct Answer: \sqrt{5}

Deg^°

Rad

Goal #4:

I can solve for the zeroes of a function using the quadratic formula.

Multiple Choice

Solve using the Quadratic Formula:

$$-3x^2+7x+5$$

• $$x\ =\ \frac{-7\pm\sqrt[]{109}}{-6}$$

• $$x\ =\frac{3\pm i\sqrt[]{131}}{14}$$

• $$x\ =\ \frac{-5\pm\sqrt[]{109}}{-6}$$

Multiple Choice

Solve using the Quadratic Formula:

$$2x^2-14x+20$$

• $$x\ =\ \left\{2,\ 5\right\}$$

• $$x\ =\frac{-2\pm2\ \sqrt[]{281}}{28}$$

• $$x\ =\ -5\pm4\ \sqrt[]{2}$$

Multiple Choice

Solve using the Quadratic Formula:

$$x^2+4x+9$$

• $$x\ =\ \frac{9\pm\sqrt[]{65}}{2}$$

• $$x\ =\frac{-1\pm\sqrt[]{145}}{8}$$

• $$x\ =\ -2\pm i\ \sqrt[]{5}$$