Wednesday Feb 21 Multiplying monomials, binomials, and beyond!

Multiplying monomials binomials and beyond

By OLIVIA GREEN

Multiplying Monomials

​What is a monomial?

Multiple Select

Identify all monomials.

• a⁷ - 3a + 1

• 15a⁸

• 5 - a⁸

• 7a + 5a¹²

• $$\frac{3}{4}a^2$$  

First, let's review exponents

​x² = x ∙ x​

​​x³ = x ∙ x ∙ x

x⁴ = ? ​

​So what about...

​​x² ∙ x³ = ?

​​

​

Multiple Choice

Simplify: $$d^4\cdot d^5$$  

• 2d⁴
• d⁹
• d²
• 2d

Multiple Choice

Simplify: $$n^2\ \cdot\ n^4$$  

• 2n²

• n⁸

• n⁶

• 2n⁸

Multiple Choice

Simplify: $$\left(-2\right)^3\ \cdot\ \left(-2\right)^6$$  

• $$\left(-2\right)^9$$  

• $$\left(-2\right)^{18}$$  

• $$\left(-4\right)^{18}$$  

• $$\left(4\right)^9$$  

​​What about this?

​​What about this?

Solution:​

Multiple Choice

$$6x^7\times4x^2$$  

• 10x⁹
• 24x¹⁴
• 24x⁹
• 10x¹⁴

Multiple Choice

$$\left(-7y^3\right)\left(5y^2\right)$$  

• -35y⁶
• -35y⁵
• -2y⁶
• -2y⁵

Multiple Choice

$$\left(-x^5\right)\left(y^3\right)$$  

• -x^8y
• -xy⁸
• -x⁵y³
• -(xy)⁸

Multiple Choice

$$9xy^2\cdot9x^5y^2$$  

• 81x⁶y⁴
• 18x⁶y⁶
• 9x⁶y⁴
• none of the above

Multiple Choice

$$7e^3\cdot10e^3f^5$$  

• 70e³f⁶

• 17e³f⁸

• 70e⁶f⁵

• none of the above

Multiple Choice

$$10xy^3\cdot8x^5y^3$$  

• 80xy³
• 80x⁶y³
• 80x⁶y⁶
• none of the above

Multiple Choice

$$\left(-4a\right)\left(3abc\right)$$  

• -12abc
• -12bc
• -4a²bc
• -12a²bc

Adding Polynomials--a quick review

If the polynomials are added, just drop the parenthesis and combine like terms.

(3x² -2x + 5) + (5x² +4x + 6) =

3x² -2x + 5 + 5x² + 4x + 6 =

8x² +2x + 11

Multiple Choice

Add the polynomials.
(3x³ + 3x² – 4x + 5) + (x³ – 2x² + x – 4)

• 4x³ + 1x² – 3x + 1
• 2x³ + 1x² - 5x + 9
• 6x³ + 1x - 1x² - 3
• 3x⁵ +1x - 1x⁵ - 3x

Subtracting Polynomials--a quick review

To Subtract Polynomials, you must distribute the subtraction sign, then combine like terms.

(4x² - 8) - (3x² - 12) the subtraction sign goes with each term in the ( )

4x² - 8 - (3x²) - (-12)=

4x² -8 -3x² +12 = 4x² -3x² -8 +12

x² + 4

Multiple Choice

(3x + 4y - 3z) - (2x - 6y + 7z)

• x + 10y - 10z
• -x +10y - 10z
• x -10y - 10z
• -x -10y - 10z

Multiplying two monomials, a quick recap!

• Multiply numbers

• Add exponents of like variables

Multiple Choice

3a(5a² + 8a + 2)

• 15a³ + 24a² + 6a
• 10a³ + 24a + 6
• 8a³ + 11a² + 6a
• 15a³ + 24a + 6

25-1 Multiplying Binomials

Two Methods: FOIL Method & BOX Method​

​

First

Outer

Inner

Last​

Multiplying Polynomials & Special Products

Multiple Choice

According to exponent rules, when we multiply terms with the same base we _______ the exponents.

• Multiply

• Add

• Divide

• Subtract

Multiple Choice

Simplify:

$$5\left(3x+2\right)$$  

• $$3x+10$$

• $$15x+10$$  

• $$15x-10$$  

• $$3x-10$$  

Multiple Choice

$$\left(x\ +\ 3\right)\left(x\ +\ 2\right)$$  

• $$x^2\ +\ 5x\ +\ 6$$  

• $$x^2\ +\ 6x\ +\ 5$$  

• $$x^2\ +x\ +5$$  

• $$x^2\ +\ 6x\ +\ 6$$  

​Box Method

MULTIPLYING LARGER POLYNOMIALS

Multiplying larger Polynomials

• When multiplying polynomials, you will be using the distributive property

• Remember how the distributive property works: when an expression is being multiplying by a term, distribute, or multiply, every term in the expression by the term

• (3)(5x + 5) = 3 (5x) + 3(5)

Multiplying larger Polynomials by the Box Method

• Create a box divided into columns and rows equal to the number of terms in each polynomial

• Write one polynomial across the top of the box, with each term corresponding to a cell. Put the sign of the term in the cell with the polynomial.

• Write the other polynomial down the side of the box, with each term corresponding to a cell.

• Put the sign of the term next to the term, not above

Multiplying larger Polynomials by the Box Method

• Multiply the upper polynomial by the first term of the side polynomial and write it in the cell underneath the term

• Multiply the upper polynomial by the second term of the side polynomial and write it in the cell underneath the term

• Continue until you have multiplied the top polynomial by every term in the side polynomial

• Simplifying by adding like terms.

• Notice the like terms follow a diagonal line

Multiplying Polynomials: Rainbow or Distributive Method

• 1. Called the rainbow method because of the lines of distribution

• 2. Multiply everything in the second polynomial by the first term in the first polynomial (along with the sign)

• 3. Multiply everything in the second polynomial by the second term (along with the sign)

Multiplying Polynomials: Rainbow or Distributive Method

• 4. Continue the process until there are no more terms in the first polynomial to multiply by

• 5. Simplify by combining like terms.

Multiple Choice

Multiply.
(5x + 2)(x² - 3x + 6)

• 5x³ - 17x² +24x +12
• 5x³ + 17x² - 24x +12
• 5x³ - 13x² + 24x +12
• 5x³ +13x² - 24x - 12

Multiple Choice

Multiply.
(b + 3)(b² + 2b + 1)

• b³+6b²+6b+3
• b³+5b²+7b+3
• 4b³+8b²+3b
• 3b³+6b²+3b

Multiple Choice

CHallenge Problem

$$\left(5s+2\right)^2$$  

• $$25s^2+4$$  

• $$25s^2+10s+4$$

• $$25s^2+20s+4$$  

• $$5s^2+20s+4$$  

Multiple Choice

$$3x^2\left(2x+6x^2+2\right)$$  

• $$6x^1+10x^4+2x^2$$  

• $$2x^3+3x^4+2x^2$$  

• $$6x^3+18x^4+6x^2$$  

Multiple Choice

$$\left(4x+3\right)\left(2x+1\right)$$  

• $$8x^2+10x+3$$  

• $$8x^2+3$$  

• $$6x^2+10x+3$$  

• $$6x^2+3$$