Polynomials-Characteristics and Operations

Polynomials-Operations

Review Polynomials---what they are and how to add-subtract-multiply and divide them

The purpose of this lesson:

• Identify types of Polynomials and find their degree

• Add, subtract, multiply, and divide them

• This is a review from Algebra 1!

Let's review the basics:

• writing polynomials in standard form

• finding and naming polynomials by degree

• naming polynomials by number of terms

• identifying leading coefficient and constant

Writing answers in standard form:

Write each monomial in descending exponent order. The highest exponent will be first & the constant term (the one without a variable) is last.

Look at the examples====>

Multiple Choice

Write the polynomial in standard form:    

 $8x+4x^2-5x^6$        

•  $4x^2+8x-5x^6$  

•  $-5x^6+8x+4x^2$  

•  $-5x^6+4x^2+8x$  

Multiple Choice

Write the polynomial in standard form:

 $6\ +\ 2x\ +\ 3x^2$  

•  $2x\ +\ 6\ +\ 3x^2$  

•  $3x^2\ +\ 6\ +\ 2x$  

•  $6\ +\ 2x\ +\ 3x^2$  

•  $3x^2\ +\ 2x\ +\ 6$  

Degree of a Polynomial

The 'Degree' is determined by finding the degree of each monomial and then choosing the highest value.

Degree & Name of a polynomial

Degree: Write the polynomial in standard from first, the degree of the polynomial will be the degree of the 1st term (has the highest exponent value)!

Study the chart for degree & common names ====>

Multiple Choice

What is the degree and name of the polynomial?

x²-2x+5

• 1; Linear

• 2; Quadratic

• 3; Cube

• 0; Constant

Multiple Choice

What is the degree of the polynomial?

 $3x^7y^2-5x^3y^8+2x^2yz^4$ 

Hint: find the degree of each term 1st!  What's the highest value?

• 8

• 9

• 11

• 7

Classify by number of terms:

You probably recognize these!

Study the chart ====>

Multiple Choice

Classify by number of terms:

x² + 4x -8

• binomial

• trinomial

• monomial

• polynomial

Multiple Choice

Classify by number of terms:

2n³

• Monomial

• Binomial

• Trinomial

• Polynomial

Multiple Choice

Classify by degree and number of terms:

3x² – 8x + 1

• Monomial Cubic

• Binomial Quadratic

• Trinomial Quadratic

• 4-Term Cubic Polynomial

Leading Coefficient & Constant Term

Look at the chart ===>

The leading coefficient is the number in front of the highest degree variable.

The constant is the number without a variable.

Write in standard form 1st to make these easy to identify!

The CONSTANT here is 0!

Multiple Choice

Put in standard form first. Then, give the Leading Coefficient (LC) and the Constant:

 $3x+5x^4-9$  

• LC = 3;  
  
  constant = 9

• LC = 5; 
  
  constant = 9
  

• LC = 5; 
  
  constant = -9

• LC = 4; 
  
  constant = -9

Multiple Choice

What is the leading coefficient (LC) and the constant?

f(x)=5x²+3x³-2x

• LC: 5

  Constant: 0

• LC: 3

  Constant: -2

• LC: -2

  Constant: 3

• LC: 5

  Constant: -2

Adding Polynomials

Add like terms!

Like terms have:

Same variables with same exponents!

You can add like terms vertically!

Some find this an easier way to get their answer.

Like-terms are written under each other and then added down.

Multiple Choice

Find the sum.
(2x² + 5x - 7) + ( 3 - 4x² + 6x)

• 2x² + 3x +1
• -2x² - 11x -4
• 2x² + 5x -7
• -2x² + 11x -4

Multiple Choice

Find the sum. 
(3 - 2x + 2x²) + (4x - 5 + 3x²)

• 7x - 7x + 5x²
• 5x² + 2x - 2
• 5x²
• 5x² + 6x + 8

Subtracting Polynomials

Distribute the negative and rewrite the 2nd polynomial.

Then lose the parentheses and combine like terms.

You can also line them up vertically and add down!

DISTRIBUTING THE (-) IS THE SAME AS MULTIPLYING BY -1 !

Here is the same problem done vertically

Distribute the (-) by multiplying each term by -1.  Then add down to get:

 $2x^2-6x+11$  

Multiple Choice

Find the difference. 
(3- 2x + 2x²) - (4x -5 +3x²)

• x_² + 6x + 8
• 2x² + 5x - 7
• -x² -6x + 8
• -2x² + 11x - 4

Multiple Choice

Find the difference:

 $\left(3x^2\ +\ 9\right)\ -\ \left(2x^3\ -\ 6\right)$  

•  $x^2\ +\ 3$  

•  $x^2\ +\ 15$  

•  $-2x^{3^{\ }}\ +\ 3x^2\ +\ 15$  

•  $-2x^3\ +\ 3x^2\ +\ 3$  

Multiplying

Polynomials

Here are the types of problems you'll need to be able to answer

To multiply by a MONOMIAL, just distribute

Binomial x Binomial

Distribute twice using FOIL or a

BOX. Here is the Box method ==>

Put the terms on the outside of a box and multiply. Add like terms on the diagonal.

You might have done this in Biology with Punnett Squares!

Binomial x Trinomial

Distribute each term in one expression.

Draw a 2 x 3 box to use the Box method.

Remember to write your answer in standard form!

Multiple Choice

Multiply: 2x³(x²−2x − 3)

• 2x⁶- 4x³ - 6x³

• 2x⁵−4x⁴ − 6x³

• -2x⁵−4x⁴ − 6x³

• -2x⁶−4x³ − 6x³

Multiple Choice

Multiply: (x − 3)(6x − 2)

• 6x² − 20x + 6

• 6x² + 20x - 6

• 6x² +6

• 6x + 6

Multiple Choice

Multiply: (x+10)(x-10)

• x² - 100

• x² + 100

• x² + 20x - 100

• x² -20x + 100

Multiple Choice

Multiply: (2n + 2)(6n + 1)

• 12n + 2

• 12n² + 2

• 12n² + 12n + 2

• 12n² + 14n + 2

Multiple Choice

Multiply: (4a + 2)(6a² − a + 2)

• 24a³ + 16a² + 6a + 4

• 24a³ + 8a² + 10a + 4

• 24a³ + 8a² + 6a + 4

• 24a³ - 8a² - 6a + 4

Binomial Squared?

Write the binomial TWICE and multiply (FOIL or Box).

Do NOT square each term!!!

Notice you get a trinomial.

Multiple Choice

Multiply: (4p − 1)(4p - 1)

• 16p² + 1

• 16p² − 1

• 16p² − 8p + 1

• 16p² + 8p + 1

Multiple Choice

Multiply: (2x+3)²

• 2x+3+2x+3

• 4x²+12x+9

• 4x²+9

• 16x²+9

Multiply 3 binomials:

FOIL the first 2. Multiply the result by the last binomial.

The Box method is shown for the last steps. ====>

Here's another example:

DIVIDING by a monomial

Divide each term by the monomial on the bottom. Then reduce each fraction.

Another example:

Each term on top is divided by x.

Reduce the fractions by subtracting the exponents!

Multiple Choice

Divide the polynomial by the monomial.

• 2x⁸-8x²-28x

• 2x⁸-2x²-7x

• 8x⁷-8x-28x

• 2x⁷ -2x -7

Multiple Choice

Divide the polynomial by the monomial.

• 15x⁹ - 8x³

• 3x⁶ - 8x³

• -5x¹²

• 3x⁶ - 40x³

Thanks for completing the lesson!

• Now go practice your skills!