Unit 3 Part 2: Adding/Subtracting/Multiplying Polynomials

Name: ______

3-6 Notes

Monomial
Polynomial
Binomial / Trinomial / n-terms

Like Terms: ______

Ex. Simplify each expression. Classify the polynomial.

1) 3x2 + 4x + x22) 4y2 + 3 – 2 + 5y – y2

3) -2x + 1xy + 4x – 5xy4) 4x2y – 2x + 5x2y + 2x

5) (2x2 + 2x – 4) + (x2 + 3x + 7)6) n2 + 4n – 3

+ 3n2– 4n – 2

Arranging Polynomials

Ascending Order - ______

Descending Order - ______

A. Arrange the terms of each polynomial so that the powers of x are in ascending order.

1.x4 + x3 +x22.4xy + 2y + 5x2

3.-4nx – 5n3x3 + 54.2x3 – x + 3x7

B. Arrange the terms of each polynomial so that the powers of x are in descending order.

5.ax2 + 8a2x5 – 46.20x – 10x2 + 5x3

7.9bx + 3bx2 – 6x38.x5 + x2 – x3

C. Simplify each expression. Put your answer in descending order with respect to x.

9. 2x – 3x2y + 4x2y – 4x 10. 6x2y + 3x3y3 – 4x2y + 2x

3-7 Notes

Warm-Up: Make each subtraction problem into an addition problem.

1.3 – 7 2.-5 – 6 3.1 – (-7)

To subtract polynomials: Turn into an ______problem.

Find each difference.

Method 1:Subtract in column form. Line up like terms.

1.(4x2 – 3a + 4) – (x2 + 6a + 1)

Method 2: Group like terms.

2.(3x2 + 5) – (-4x + 2x2 + 3)

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You try! Find each difference.

1.6m2 + 72.(10x2 + 8x – 6) – (3x2 + 2x – 9)

(–)-2m2 + 2m – 3

Did you hear about

3-6/3-7 Keystone Examples

Warm-Up: Simplify each expression.

1) (1 + 4n + 6n2) + (2n3 + 3n2 + n)2) (x2 – 8x4 – 8x) – (4x2 – 8x + 6x4)

Keystone Questions!

1)Give an expression to represent the perimeter of the triangle.

Perimeter: ______

2)A triangle has a perimeter of 4x2 – 3x + 1. Two of the sides are 4x + 1 and 2x2 + x – 3. Find an expression to represent the third side of the triangle.

3)The polynomial expression (my2 – 4y + 3) – (5y2 – 2y – p) is simplified to 4y2 – 2y – 4. What are the values of m and p?

Practice

1) The polynomial expression (3 – 4x + mx2) + (2x2 – 3x + n) is simplified to 6x2 – 7x + 8. What are the values of m and n?

2) The perimeter of a triangle is 19x – 7. Two sides of the triangle are represented by the expressions 5x – 9 and 8x + 3, respectively. Find the expression that represents the length of the third side.

3) A square has a side length equal to 5x – 3y inches. What is the perimeter of the square?

4) Simplify. (y2 – 5y + 3) – (3y3 + 4y2 – 6y – 10).

5) What are the values of m and n?

a)(3x2 – 2x + 4) – (2x2 + mx – n ) = x2 + 4x – 8

b)(-4x2 + 6x – m) – (3x2 + nx – 4) = -7x2 + 3x – 8

c)4x2 – 7x + 3 + mx2 – nx + 6 = -2x2 – 3x + 9

Let’s Review what we’ve learned in Unit 3 so far!

Simplify each exponential function.

1) x2(x3)2) m(m2) 3) 3x2(2x)

4) 5) (3x2)36)

Simplify each expression. Write your answer in descending order.

7) (3x3 – 2x2 + 4) + (-2x2 – 3) 8) (2x2 – 4x) – (2x2 + 4)

9) (7x – 2x2 + 10) – (-5x2 + 2x – 18) 10) 10x2y– 4xy2 + 3x2y – 3xy2

11) x2 – 4x4 + 3 + 9x2 – 2x4 – 1 12) (-2x2 – 3xy) – (4x2 + 2xy)

13) Give an expression to represent the perimeter of the rectangle.

x2 + 3

3x – 2

3-8 NotesMultiplying a polynomial by a monomial

Consider the this expression: 6x(x – 3)6x is a ______

(x – 3) is a ______

These two terms are being ______.

*This is the ______property!

Examples:

1. 6(x + 7)

2. -2(g2 + 3g – 5)

3.9a(a + 1)

4.2y2(3y2 + 2y – 7)

5.-3x2(x3 – 2x2 + 3)

6.4g(5g2 – 3w + 2)

3-8 Practice

Find each product.

1.8(x + 2)2.-3(x + 5)

3.10(2a – b)4.4v(v2 – 4v + 9)

5.-3y(y – 1)6.2r(-2r2 + 6r – 5)

7.0.3(2p + 4)8.4n2(m3 + m2)

9.½(2z2 + 4z + 10)10.4x(y + 6)

11.8a(a – 2b)12.-5m2(m + 3)

13.-7d(d3 – 2)14.h(h + 4)

15.k(k – 9)16.6p2(p – 8)

3-9 Notes Multiplying Binomials and Polynomials

A.Multiplying Two Binomials

*Each term in the ____ binomial has to multiply each term in the _____ binomial.

F O I L

Examples: Find each product

1.(m – 3)(m + 4)2.(3a + 11)(5a – 2)

You try!(4x + 7)(3x – 8)

Multiplying Trinomials

3.(2x + 3)(x2 + 3x + 8)4.(2y + 5)(3y2 – 8y + 7)

3-9 Practice

Find each product.

1.(x + 2)(x + 3)2.(x – 4)(x + 1)3.(x – 6)(x – 2)

4.(p – 4)(p + 2)5.(y + 5)(y + 2)6.(2x – 1)(x + 5)

7.(3n – 4)(3n – 4)8.(8m – 2)(8m + 2)9.(k + 4)(5k – 1)

10.(x + 2)(x2 – 2x + 1)11.(x + 3)(2x2 + x – 3)

12.(2x – 1)(x2 – x + 2)13.(p – 3)(p2 – 4p + 2)

3-10 Notes

Special Products

Recall: 32 means ______

(Binomial)2 means ______

1.(3a + 2)2

2.(6p + 11q)2

3.(y – 13)2

4.(4x2 – 7t)2

5.(6x – 20y)(6x + 20y)

6.(14k + 9t2)(14k – 9t2)

3-10study guide

Keystone Questions!

Ex 1) a. Find the area of the shaded region in simplest form.

b. What is the area of the shaded region if x = 3 inches?

Ex 2) Conner tosses a dog treat upward for his dog to catch. The height of the treat in the air in meters, h, after t seconds is given by the equation h = 13.7t – 4.9t2. How high is the treat after 1.5 seconds? Show all of your work.

Height of dog treat: ______

Ex 3) The perimeter of the triangle below is P = 5x2 – 2x + 3. Find an expression that represents the measure of the third side.

You Try!

  1. You plan to build a house that is 1.5 times as long as it is wide. You want the land around the house to be 20 feet wider than the width of the house, and two times as long as the length of the house, as shown below.
  1. Write an expression for the area of the land surrounding the house.
  1. If x = 30 feet, what is the area of the land surrounding the house?
  1. Since 2006, the number of Kendamas, K (in thousands) produced at a manufacturing plant can be modeled by the equation K = 3t2 – 2t + 10, where t is the number of years since 2006. Use this equation to predict how many Kendamas will be made in the year 2014. Show all of your work.

Number of Kendamas = ______

  1. Find the area of the shaded region.