Add, Subtract, and Multiply Polynomials

Add, Subtract, Multiply Polynomials EDG 2.5-6

CLASSWORK GRADE

The purpose of this lesson:

• You will review/practice how to add, subtract, and multiply polynomials. You did most of this in Algebra 1.

• The grade you make will be a daily grade.

• This assignment will replace Edgenuity 2.5-6

Before we Add, Subtract, and Multiply; Let's review the following

• writing polynomials in standard form

• naming polynomials by degree

• naming polynomials by number of terms

• identifying leading coefficient and constant terms

Answers should be written in standard form

Put exponents in descending order. The highest exponent will be first & the term without a variable is last.

Look at these examples ----->

Multiple Choice

Put this polynomial in standard form:

$$7x+2x^2-3x^4$$  

• $$2x^2+7x-3x^4$$  

• $$-3x^4+2x^2+7x$$  

• $$-3x^4+7x+2x^2$$  

Multiple Choice

What is the standard form of the polynomial,
 9x² + 5x + 27 + 2x³

• 27 + 5x + 9x² + 2x³
• 9x² + 5x + 2x³ + 27
• 9x² + 5x + 27 + 2x³
• 2x³ + 9x² + 5x + 27

Degree of a Polynomial

the 'degree' is determined by the highest exponent. It does not matter how many terms the polynomial has.

STUDY THE CHART ON THE NEXT SLIDE.

YOU LEARNED THIS VOCAB. IN ALGEBRA 1

Multiple Choice

Classify by the DEGREE:

4x³ - 5x² + 2x - 1

• 1; Linear

• 2; Quadratic

• 3; Cubic

• 4; Quartic

Multiple Choice

What is the degree of the polynomial 17x + 4

• 0; Constant

• 1; Linear

• 2; Quadratic

• 3; Cubic

Multiple Choice

What is the degree of the polynomial

x²-2x+5

• 1; Linear

• 2; Quadratic

• 3; Cubic

• 0; Constant

Classify by number of terms:

You already know this!

Study the chart ------->

Multiple Choice

Classify:
x³ + 3x

• Monomial
• Binomial
• Trinomial
• Polynomial

Multiple Choice

Classify by DEGREE and TERMS:

3x² – 8x + 1

• Cubic Monomial

• Quadratic Binomial

• Quadratic Trinomial

• Cubic Polynomial

Multiple Choice

Classify by DEGREE and TERMS:

3x³ – 6x

• Linear Monomial

• Cubic Binomial

• Quadratic Trinomial

• Linear Polynomial

LEADING COEFFICIENT AND CONSTANT TERM

Study the chart ------>

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

The constant is the number without a variable.

The CONSTANT in this example is 0

Multiple Choice

Put in standard form. Then, tell the lead coefficient (LC) and the constant: 3x+5x⁴ - 8

• LC: 3

  constant: 8

• LC: 4

  constant -8

• LC: 5

  constant 4

• LC: 5

  constant: -8

Multiple Choice

Tell the lead coefficient (LC) and the constant:

-3x² + 8x

• LC: -3

  constant: 0

• LC: 3

  constant: 8

• LC: 2

  constant: 0

• LC: -3

  constant: 8

ADDING POLYNOMIALS

Just combine like-terms

Like terms must have the exact same variables and exponents

You can combine like-terms vertically as well ----->

Some people like to show their work in this way.

Multiple Choice

Add the polynomials.

(2xy - 3x² + 4y - 5) + (-3xy + 4y² - 3y + 8)

• -1xy - 3x² + 4y² + 1y + 3

• -1xy + 1x² + 1y + 3

• -1xy + 1y² + 1y + 3

• 1xy + 3x² + 4y² + 1y + 3

Multiple Choice

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

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

Adding polynomials to find perimeter

The perimeter of rectangle ABCD is (4x+8) + (4x+8) + (2x + 3) + (2x + 3) =

12x + 22

You need to label and add ALL sides

For the next 2 problems, remember you can click on the picture to make it larger.

REMEMBER TO ADD UP ALL SIDES

Multiple Choice

What is the perimeter of the rectangle?

• x²+8x-5

• 2x²+16x-10

• 2x²+10x-8

• 6x-2

Multiple Choice

Find the perimeter of the triangle.

• 7x + 5

• 24x + 4

• 9x + 5

• 9x² + 5

The next problem is more challenging. Read carefully. Maybe draw a picture.

Multiple Choice

The measure of the perimeter of a triangle is 37s + 42. It is known that two of the sides of the triangle have measures of 14s + 16 and 10s + 20. Find the length of the third side.

• 13s + 6

• 61s + 78

• 24s + 36

• 129s

SUBTRACTING POLYNOMIALS

Distribute and Re-write the second polynomial. After you distribute, you can lose parenthesis and combine like-terms.

DISTRIBUTING THE (-) IS LIKE MULTIPLYING BY -1

Here is the same problem done vertically

Distribute -1 and combine like terms vertically:

 $2x^2-6x+11$  

Multiple Choice

Find the Difference.
(4x³ -3x² + 6x - 4) - (-2x³ + x² -20)

• 4x² - 6x³ - 2x + 6
• 3x + 5x² + 1
• 6X³ -4X² + 6X + 16
• 3x + 2

Multiple Choice

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

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

Multiplying Polynomials

To multiply by a MONOMIAL, just distribute

Binomial x Binomial

Use FOIL or the box method

Binomial x Trinomial

Distribute 6 times. You can draw a

2 x 3 box if you want to.

Remember to answer in standard form.

Multiple Choice

5x(2x³+6)

• 10x⁴+30x
• 10x³+30x
• 7x⁴+11x

Multiple Choice

(2x + y)(x +4y)

• 2x² + 8xy + 4y²
• 2x² + 9xy + 4y²
• 2x² + 4y²
• 2x + 9xy + 4y

Multiple Choice

(4x + 3)(x + 2)

• x² + 11x + 6
• 4x² + 8x + 6
• 4x² + 3x + 6
• 4x² + 11x + 6

Multiple Choice

Find the product:
(5n-8)(5n+8)

• 25n²-64
• 10n²
• 5n²+16
• 16

Multiple Choice

Multiply:

$$\left(2x-7\right)\left(x+2\right)$$  

• $$2x^2-5x+4$$  

• $$2x^2+3x+14$$  

• $$2x^2-3x-14$$  

• $$x^2+3x-14$$  

Multiple Choice

Multiply:

(5x + 2) (x² – 3x + 6)

• 5x³ – 17x² + 24x + 12

• 5x³ + 17x² – 24x + 12

• 5x³ – 13x² + 24x + 12

• 5x³ + 13x² – 24x – 12

Area Models

When you use a box to multiply, it can be called an 'area model'. You can apply this to finding the area of a rectangle -------->

MULTIPLY: LENGTH X WIDTH

Multiple Choice

What is the AREA for the given rectangle:

• 6x³ - 9x² + 10x - 15

• 6x² + 1x - 15

• 6x² + 4x + 4

• 6x³ + 1x² - 15x

Square a Binomial

Write it twice and FOIL

Multiple Choice

Find the product:
(2x+3)²

• 2x+3+2x+3
• 4x²+12x+9
• 4x²+9
• 16x²+9

Multiple Choice

(a+b)(a+b)

• a² - b²
• a² + b²
• a² + ab + b²
• a² + 2ab + b²

Exponent Rules

When multiplying like-bases, you add the exponents. Study the example ------>

MORE EXAMPLES!

Multiple Choice

(2n⁵p³)(5n⁴p)

• 7n²⁰p³

• 10n²⁰p³

• 10n⁹p⁴

• none of the above

Multiple Choice

4m(3m⁷ + 2m⁵)

• 7m⁷ + 6m⁶

• 12m⁸ + 8m⁶

• 12m⁷ + 8m⁵

How to multiply 3 binomials:

Foil the first 2. Then, multiply the resulting trinomial by the last binomial. See example --->

Another Example:

Multiple Choice

Find the volume of the rectangular prism with length ( x + 2) width

( x - 4) and height ( 2x + 1). (HINT: V = lwh)

• x² -8x

• x² -2x -8

• 2x³-3x² -18x-8

• 2x³ -8

Multiple Choice

What is the volume of the rectangular prism? (multiply length x width x height)

• 7x³ + 20x² - 10x + 6

• 6x³ + 22x² + 10x - 6

• 6x³ - 22x² - 10x - 6

• 18x² + 10x - 6

What to do next:

The grade from this assignment will be a classwork grade.

Next, you will complete the practice in your packet.

Then, you will be ready to take the Quiz. It will be posted in Google Classroom.