Algebra 2 Study Guide

Synthetic Division

What is synthetic division?

• Shortcut way to divide polynomials

• Uses the roots (not factor form) as the divisor

• Uses the coefficients of the terms, and not the variables, as the dividend

Steps to dividing by synthetic division

• 1. Change the factor form of the divisor to the root. (factor form is (x - 3), root is 3)

• 2. Write out the polynomial to be divided in descending form, writing in zeros for the terms that are missing. For example: x⁴ + 2x² + 4 would be written as: x⁴ + 0x³ + 2x² + 0x + 4

Steps to dividing by synthetic division (cont)

• 3. Write out the coefficients only of the polynomial, including signs and including the 0's

• 4. Write the root to the left of the coefficients with a vertical bar between it and the coefficients and a horizontal line underneath the coefficients (like an upside-down division symbol)

Steps to dividing by synthetic division (cont)

• 5. Bring the first coefficient down below the line, without doing anything to it

• 6. Multiply the coefficient by the root, and add the number to the next term.

• 7. Multiply the sum by the coefficient and add to the next term.

Steps to dividing by synthetic division (cont)

• Repeat this process until there are no more terms.

• If the remainder is 0, then the factor is a true root

• If the remainder is not 0, then the value of the remainder is the value of the polynomial evaluated at the x-value of the divisor (what you would get if you substituted the divisor into the polynomial)

Steps to dividing by synthetic division (cont)

• Note: You can only use this form if your divisor is a binomial

• Always remember to put in the missing terms with a zero coefficient

• The remaining number will start with a variable exponent of one less than the original. In the example to the right, the polynomial was 3rd degree, the quotient is 2nd degree.

​In this lesson we are going to divide this polynomial by this binomial.

Multiple Choice

What are the coefficients we use?

• 5, 6, 28, 2

• 5, -6, -28, -2

• -5, 6, 28, 2

• -5, -6, -28, -2

Multiple Choice

What number do we put on the outside?

• -2

• 2

• -5

• 5

Multiple Choice

Is this the way to start?

• No, it should be a positive 2 on the outside.

• No, the coefficients are in the wrong order.

• yes

Multiple Choice

What is the first thing we do?

• Bring down the 5 below the line.

• Add all the coefficients

• Divide 5 by -2

• Multiply the 5 and 2

Multiple Choice

What goes next to the 5 below the line?

• 4

• -16

• 16

• -4

Multiple Choice

What goes under the -28?

• 32

• -32

• -20

• 20

Multiple Choice

What goes next to the 32?

• -4

• 4

• 8

• -8

Multiple Choice

What is the final answer?

• $$5x^{4\ }-16x^3+4x^2-10x$$

• $$5x^3-16x^2+4x-10$$

• $$5x^2-16x+4+\frac{10}{x+2}$$

• $$5x^2-16x+4-\frac{10}{x+2}$$

Multiple Choice

Simplify using synthetic division. 

$$\frac{7v^3-59v^2+71v-11}{v-7}$$  

• $$7v^2-10v+2-\frac{4}{v-7}$$  

• $$7v^2-10v+1-\frac{5}{v-7}$$  

• $$7v^2-10v+1-\frac{4}{v-7}$$  

• $$7v^2-10v-\frac{2}{v-7}$$  

Multiple Choice

Simplify using synthetic division.

$$\frac{p^3-2p^2-88p-64}{p+8}$$  

• $$p^2-10p-8$$  

• $$p^2-10p-10$$  

• $$p^2-10p-7-\frac{3}{p+8}$$  

• $$p^2-10p-11-\frac{3}{p+8}$$  

Multiple Choice

Rewrite the rational expression. 

$$\frac{-4x^3+7x^2-2x+1}{x-2}$$  

• $$-4x^2-x-5-\frac{8}{x-2}$$  

• $$-4x^2-x-4-\frac{7}{x-2}$$  

• $$-4x^2-x-3-\frac{8}{x-2}$$  

• $$-4x^2-x-7-\frac{12}{x-2}$$  

Multiple Choice

Divide using synthetic division:

$$\left(2x^2+6x-20\right)\div\left(2x-4\right)$$  

• $$x+5+\frac{2}{2x-4}$$  

• $$x-8$$  

• $$x-5+\frac{3}{2x-4}$$  

• $$x+5$$   

Multiple Choice

Divide using synthetic division

(the optional video clip will help with this problem)

  $$(4x^4-3x^3-4x^2-x+3)\ ÷\ (4x-3)$$  

• $$x^3+x^2+x-1$$

• $$x^3-x-1$$

• $$x^3-2x-1$$

• $$x^3-x^2+x-3$$

Here are the steps for polynomial long division:

Step 1: Set up the problem, Are polynomials in standard form? is a placeholder needed?

Step 2: Divide, and write something on top

Step 3: Multiply by what is outside

Step 4: Subtract

Repeat until you cannot divide any longer.

Write final answer in proper format (if there is a remainder).

***If there is no remainder, that means the divisor is a factor of the dividend BUT IF THERE IS a remainder, the divisor IS NOT A FACTOR of the dividend.

Multiple Choice

$$\frac{8x^2+6x-20}{4x-5}$$  

• $$2x+4-\frac{40}{4x-5}$$  

• $$2x+4$$  

• $$2x-1-\frac{16}{4x-5}$$  

• $$2x-4$$  

Multiple Choice

$$\frac{27x^3+9x^2-3x-10}{3x-2}$$  

• $$9x^4+5$$  

• $$9x^2-9x+5$$  

• $$9x^2+9x+5$$  

• $$9x^2-9x$$  

Multiple Choice

Dividing polynomials requires that your dividend be written both in standard form and such that no variable is missing between the largest and smallest exponent (in descending order). You must insert placeholders (zeroes) if a term(s) is missing in your dividend. Does the polynomial $$4x^2+23x-16$$  require any placeholders?

• No

• Yes

• There is not enough information.

Now that you have confirmed that your dividend is written correctly, consider what your next step should be and answer the following question.

Multiple Choice

What step should you take to begin this problem.

• Divide 23x by x and write "23" above 23x. 

• Divide  $$4x^2$$  by x and write 4x above 23x.

• Divide  $$4x^2$$  and 23x by x and then write your answer in order above the dividend.

• Multiply -16 by x and 5 and subtract your answer from the dividend. 

Multiple Choice

After dividing $$4x^2$$  by x and writing 4x above 23x, what is the next step that you should take in solving this problem?

• Divide 23x by x and write 23 above -16.

• Multiply 4x by x and write $$4x^2$$  beneath $$4x^2$$  .

• Multiply 4x by x and 5, then write $$4x^2+20$$    beneath $$4x^2+23x$$  .

• Multiply 4x by x and 5, then write $$4x^2+20$$    beneath $$4x^2+23$$  .

Multiple Choice

The next step you should take is to "subtract." What difference will you find?

• $$0x^2+3x$$  

• $$8x^2+43x$$  

• $$8x^2+3x$$  

• $$0x^2-3x$$  

Multiple Choice

Which selection is the final answer?

• $$4x+3-31$$  

• $$4x-28$$  

• $$4x+3-\frac{31}{4x+3}$$  

• $$4x+3-\frac{31}{x+5}$$  

​This is a separate example for you to view. Notice that the remainder is written a bit differently in this problem.

Multiple Choice

Consider the given polynomial $$x^2+3$$  . Does the polynomial require placeholders before long division can be performed? If yes, pick the correct placeholder.

• No

• Yes, 0x should be included.

• Yes, $$0x^3$$  should be included.

• Yes, $$0x^2$$  should be included.

Multiple Choice

Perform the given division problem. What is your final answer? Take your time, and show your work. This will be graded.

• $$x+1+\frac{4}{x-1}$$  

• $$x-1-\frac{4}{x-1}$$  

• $$x+1-\frac{4}{x+1}$$  

• $$x+1+\frac{3}{x-1}$$  

Let's analyze some remainders!

Let's start by using synthetic division together.

Think to yourself. What is the polynomial represented by this synthetic division? (You will answer on the next slide)

Multiple Choice

What is the remainder if $$P\left(x\right)=x^4-3x^2-2x+5$$  is divided by $$x+2$$  ?

• 13

• 23

• -13

• -23

Multiple Choice

Try it now using the remainder theorem:

What is the remainder of (2x³ - x² - 13x + 9) divided by (x - 2)

• -7

• -5

• 7

• 5

Multiple Choice

If $$\left(9x^4-45x^3+37x^2+x+2\right)$$ is divided by $$\left(x-2\right)$$ then the remainder is...

• -64

• 64

• 656

• -652

Multiple Choice

Decide if (x-3) is a factor of 3x³+10x²-x-12

• Yes, it is a factor!
• No, it is not a factor!

Multiple Choice

Is (x-4) a factor of (x³ +x² -16x-16)?

• YES
• NO

​Example 1: Use Polynomial Long Division to find the remainder of the problem below. Verify using the Remainder Theorem.

​Example 2: Using the Remainder Theorem, find the remainder of when

​Therefore, the remainder is -13

Let’s check it out:

f(-1) = 3(-1)3+(-1)2+2(-1)+5
f(-1) = -3 + 1 -2 + 5
f(-1) = 1

If there is a
remainder then (-1) is
not a zero of the
polynomial.

Multiple Choice

Try it now using the remainder theorem:

What is the remainder of (2x³ - x² - 13x + 9) divided by (x - 2)

• -7

• -5

• 7

• 5

FACTOR THEOREM

​Example Question: Is x+1 a factor of p(x) = x¹¹ - 4x - 3?

Answer:

If p(-1) = 0, then the remainder must be zero when p(x) is divided by (x+1), which would mean (x+1) is a factor. So first evaluate p(-1).

p(-1) = (-1)¹¹ - 4(-1) - 3

p(-1) = -1 + 4 - 3

p(-1) = 0

x+1 is a factor of p(x)!

FACTOR THEOREM

Multiple Choice

You Try!

Is $$x-3$$   a factor of $$x^5+16x-200$$  ?

• Yes, because when we evaluate the polynomial for x = 3, we get 0.

• Yes, because when we evaluate the polynomial for x = 3, we get 3.

• No, because when we evaluate the polynomial for x = 3, we get 91.

• No, because when we evaluate the polynomial for x = 3, we get -491.

Multiple Choice

Is $$x-3$$  a factor of $$3-7x+5x^2-x^3$$  ?

• Yes

• No

Multiple Choice

Is $$2x-1$$  a factor of $$2x^3+3x^2-8x+3$$  ?

• Yes

• No

Polynomials

Vocabulary, Classifications, and Operations

Polynomials Vocabulary

● Variable (a letter) - represents a quantity that can

vary or change

● Constant (not changing fixed) - a number or symbol

for a non-changing value (6, 234, 𝜋 ) that is NOT
next to a letter (not being multiplied by a variable)

● Coefficient - a number or value NEXT TO A

LETTER or variable that acts as a multiplier
(including the invisible 1)

Polynomials Vocabulary

● Monomial - a number, a variable, or the product

of a number and one or more variables (1 term)

● Binomial - polynomial with 2 terms
● Trinomial - polynomial with 3 terms
● Polynomial - a monomial or the sum of

monomials (4 terms or more)

In your notebook, write the expression below and label it
(like in the examples above) with these vocabulary words:
Constant, Variable, Coefficient, Exponent, Base, Term.

Multiple Choice

Classify the following polynomial

• quartic polynomial

• quadratic polynomial

• quartic trinomial

• quadratic trinomial

Multiple Choice

Classify the following polynomial

• quadratic trinomial
• cubic binomial

Multiple Choice

Classify the polynomial:
3x² – 8x + 1

• quadratic trinomial

• cubic trinomial

• quadratic binomial

• cubic binomial 

​TERM

- a number, a variable, or the product of a number and variable(s)

​EXAMPLES

​​poly- "many" -nomial "term" Term(s) whose exponents are whole numbers, and are separated by a "+" or a "-"

​POLYNOMIAL

​​x + 3

​​x² - 4x - 9

​9​x³ + 5x² - 3x + 12

​How many terms does each polynomial have?

Math Response

How many terms does the polynomial have?

$$2ab^2-4c+6d^3-12ef\ +\ 101$$  

Type answer here

Correct Answer: 5

Deg^°

Rad

​STANDARD FORM of a POLYNOMIAL

all of the terms are in order from the highest exponent (degree) to the lowest exponent (degree).

​EXAMPLE

Write in STANDARD FORM

​5y² + w⁴ + w⁷ - q³ + 2y - 10

Reorder

Rewrite $$2x^2+9-5x^3+3x^4-8x$$ in standard form.

• $$+3x^4$$

• $$-5x^3$$

• $$+2x^2$$

• $$-8x$$

• $$9$$

1

2

3

4

5

• 1
• 2
• 3
• 4
• 5

​DEGREE OF A POLYNOMIAL

​EXAMPLES

​3x² + 9x - 4

​x² - 8x⁵ + 7

​y⁵z⁴ + w²x³ + 2xy - 10

​the degree is ___

​the degree is ___

​the degree is ___

the highest degree of its monomials with non-zero coefficients.

Math Response

What is the degree of the polynomial?

$$a^3b^3+c^4d-x^3y^2$$  

Type answer here

Correct Answer: 6

Deg^°

Rad

​LEAD COEFFICIENT

the coefficient of the first term when the polynomial is written in standard form.

​EXAMPLE

What is the lead coefficient?

​-9x³ + 3x - 6x⁴ + 22

Math Response

What is the lead coefficient of the polynomial?

$$-4a^3+19a^4-a+12a^6+103$$  

Type answer here

Correct Answer: 12

Deg^°

Rad

​NAMING A POLYNOMIAL

​by the number its degree

​DEGREE	NAME​	​EXAMPLE
0​	CONSTANT​	12​
1​	LINEAR​	​3x - 12
2​	QUADRATIC​	​
3​	CUBIC​	​
4​	4th DEGREE POLYNOMIAL	​

​x² + 4x - 8

4x³ - 5​x² + 4x

​9x⁴ - 2x² - 6

Multiple Choice

Name the polynomial by its degree.

$$3x^2+7x-8$$  

• constant

• linear

• quadratic

• cubic

​NAMING A POLYNOMIAL

​by the number of terms it has

​Number of Terms	​Name	​Example
​1	MONOMIAL​	​4x
2​	BINOMIAL​	​7x - 8
3​	TRINOMIAL​	​2x - 7y + 12

Multiple Select

Chose all the descriptions that describe the polynomial.

$$-2x^2-7$$  

• trinomial

• linear

• quadratic

• binomial

• monomial

Lesson Vocabulary

3

Polynomial

7x²+ 3x + 1

Monomial

Ex: 5x, 8 , 3x²

Binomial

Ex: 4x⁴ + 9

Trinomial

Ex: 9x² + 7x − 13

Constant

Ex: 23, 6, −13

1 term

2 terms

3 terms

Contains no variables

Coefficient

Constant

Term

​1 min

Multiple Choice

What is the degree of the polynomial?

$$f\left(x\right)\ =\ 2x^4-3x^2-2x\ -4?$$  

• 4

• 3

• 2

• 1

Multiple Choice

What is the leading term of the polynomial?

$$f\left(x\right)\ =\ 3x^3-6x^5-2x^2-5\ ?$$  

• $$3x^3$$  

• $$-6x^5$$  

• $$-2x^2$$  

• -5

Multiple Choice

Is the function positive or negative from a to b?

• Positive

• Negative

Multiple Choice

Is the function positive or negative from c to d?

• Positive

• Negative

Multiple Choice

Is the function positive or negative from d to e?

• Positive

• Negative