Identifying Parts & Simplifying Algebraic Expressions (or Polynomials) WITH EXPLANATIONS!

Simplify using the Distributive Property: (16 + 12)5

(16 + 12)5 = (5 × 16) + (5 × 12) = 80 + 60 = 140 *Always distribute to the LEAD term first! This matters more with subtraction. You can also use the *Commutative Property* to place the 5 in front of the quantity. Ex. 5(16 + 12)

What are the steps to simplifying an algebraic expression?

*STEP 1:* Remove any parentheses using the Distributive Property. *STEP 2:* Combine Like Terms.

What are the constant(s) in the algebraic expression (polynomial) 4x + 5.5y + 6y - 7 − x + 3?

-7, 3

Simplify using the Distributive Property: -8(2y + 2x)

-8(2y + 2x) = (-8 ⋅ 2y) + (-8 ⋅ 2x) = -16y + (-16x) or = -16x + (-16y) or = -16x - 16y

Simplify using the Distributive Property: -8.1(3m + 9)

-8.1(3m + 9) = (-8.1 × 3m) + (-8.1 × 9) = -24.3m + (-72.9) or = -24.3m - 72.9 or = -72.9 + (-24.3m) or = -72.9 - 24.3m

Simplify using the Distributive Property: -½(2n - ¼)

-½(2n - ¼) = (-½ ⋅ 2n) - (-½ ⋅ ¼) = -1n - (-⅛) = -n - (-⅛) or = -n + ⅛ or = ⅛ + (-n) or = ⅛ - n Or, you can replace ⅛ with 0.225 in any of the above responses.

Simplify using the Distributive Property: -½(x - 8)

-½(x - 8) = (-½ ⋅ x) - (-½ ⋅ 8) = -½x - (-4) or = -½x + 4 or 4 + (-½x) or = 4 - ½x Or, you can replace -1/2 with -0.5 in any of the above responses.

Simplify using the Distributive Property: 3(4 + 1)

3(4 + 1) = (3 × 4) + (3 × 1) = 12 + 3 = 15

Simplify using the Distributive Property: 8(3 - 5)

8(3 - 5) *When the number to be distributed comes after the parentheses, still ALWAYS distribute to the first term in the parentheses (from left to right). 8(3 - 5) = (8 × 3) - (8 × 5) = 24 - 40 = 24 + (-40) = -16

Simplify using the Distributive Property: ⅓x(½x + ⅓)

⅓x(½x + ⅓) = (⅓x ⋅ ½x) + (⅓x ⋅ ⅓) = ⅙x² + ⅑x or = ⅑x + ⅙x²

Simplify using the Distributive Property: ⅕(3y + ½)

⅕(3y + ½) = (⅕ ⋅ 3y) + (⅕ ⋅ ½) = ⅗y + ⅒ or = ⅒ + ⅗y Or, you can replace ⅗ with 0.6 and ⅒ with 0.1 in any of the above responses.

Simplify using the Distributive Property: 2(4n - 7)

2(4n - 7) = (2 ⋅ 4n) - (2 ⋅ 7) = 8n - 14 or = 8n + (-14) or = -14 + 8n

What are the variable(s) in the algebraic expression (polynomial) 4x + 5.5y + 6y - 7 − x + 3?

x, y

Simplify using the Distributive Property: 0.3(60a + 35b - 22c + 1)

0.3(60a + 35b - 22c + 1) = (0.3 × 60a) + (0.3 × 35b) - (0.3 × 22c) + (0.3 × 1) = 18a + 10.5b - 6.6c + 0.3 or = 18a + 10.5b + (-6.6c) + 0.3

Simplify: 2(b + c) + 3(b + c) + 5b

2(b + c) + 3(b + c) + 5b Simplify the expression: STEP 1: Remove parentheses from left to right using the Distributive Property. Keep the "+5b" by bringing it down. 2(b + c) + 3(b + c) + 5b = (2 ⋅ b) + (2 ⋅ c) + (3 ⋅ b) + (3 ⋅ c) + 5b = 2b + 2c + 3b + 3c + 5b = 2b + 3b + 5b + 2c + 3c STEP 2: Combine like terms. Add 2b and 3b and 5b to get 10b. Add 2c and 3c to get 5c.. = 2b + 3b + 5b + 2c + 3c = (2b + 3b + 5b) + (2c + 3c) = 10b + 5c *10b + 5c is the simplified form because there are no parentheses and no like terms.

Simplify: 2(x + 3y) - 3y

2(x + 3y) - 3y Simplify the expression: STEP 1: Remove parentheses using the Distributive Property. Keep the "-3y" by bringing it down. 2(x + 3y) - 3y = (2 ⋅ x) + (2 ⋅ 3y) - 3y = 2x + 6y - 3y STEP 2: Combine like terms. Subtract 6y and 3y to get 3y. = 2x + 6y - 3y = 2x + (6y - 3y) = 2x + 3y *2x + 3y is the simplified form because there are no parentheses and no like terms.

Simplify: 2x - (3x - 4) + 5

2x - (3x - 4) + 5 Simplify the expression: STEP 1: Remove parentheses from left to right using the Distributive Property. This is a tricky one! There is a hidden -1 in front of the parentheses (because of the subtraction sign) that you need to distribute. When you multiply a number or expression by -1, it will turn the number or expression into its opposite. 2x - (3x - 4) + 5 = 2x + (-1)(3x - 4) + 5 = 2x + (-1 ⋅ 3x) - (-1 ⋅ 4) + 5 = 2x + (-3x) - (-4) + 5 = 2x + (-3x) + 4 + 5 STEP 2: Combine like terms. Combine 2x and (-3x) to get -1x or -x. Combine the constants to get 5. 2x + (-3x) + 4 + 5 = [2x + (-3x)] + (4 + 5) = -x + 9 *-x + 9 is the simplified form because there are no parentheses and no like terms.

Simplify: 2y + 3(4y + 5) + 4

2y + 3(4y + 5) + 4 Simplify the expression: STEP 1: Remove parentheses using the Distributive Property. Keep the "2y +" and the "+4" by bringing them down. 2y + 3(4y + 5) + 4 = 2y + (3 ⋅ 4y) + (3 ⋅ 5) + 4 = 2y + 12y + 15 + 4 STEP 2: Combine like terms. Add 2y and 12y to get 14y. Add 15 and 4 to get 19. = 2y + 12y + 15 + 4 = (2y + 12y) + (15 + 4) = 14y + 19 *14y + 19 is the simplified form because there are no parentheses and no like terms.

Simplify: 3 - 2(-4x + 1) - 7x

3 - 2(-4x + 1) - 7x Simplify the expression: STEP 1: Remove parentheses from left to right using the Distributive Property. This is a tricky one! There is a hidden -2 in front of the parentheses (because of the subtraction sign) that you need to distribute. 3 - 2(-4x + 1) - 7x = 3 + (-2)(-4x + 1) - 7x = 3 + (-2 ⋅ -4x) + (-2 ⋅ 1) + (-7x) = 3 + 8x + (-2) + (-7x) = 8x + (-7x) + 3 + (-2) STEP 2: Combine like terms. Combine 8x and (-7x) to get 1x or x. Combine the constants to get 1. 8x + (-7x) + 3 + (-2) = [8x + (-7x)] + [3 + (-2)] = x + 1 *x + 1 is the simplified form because there are no parentheses and no like terms.

Simplify using the Distributive Property: 3(2a + 12b)

3(2a + 12b) = (3 ⋅ 2a) + (3 ⋅ 12b) = 6a + 36b

Simplify: 3(n - 7) + 4(n + 2)

3(n - 7) + 4(n + 2) Simplify the expression: STEP 1: Remove parentheses from left to right using the Distributive Property. 3(n - 7) + 4(n + 2) = (3 ⋅ n) - (3 ⋅ 7) + (4 ⋅ n) + (4 ⋅ 2) = 3n - 21 + 4n + 8 = 3n + (-21) + 4n + 8 = 3n + 4n + (-21) + 8 STEP 2: Combine like terms. Add 3n and 4n to get 7n. Subtract the constants to get -13. = 3n + 4n + (-21) + 8 = (3n + 4n) + [(-21) + 8] = 7n + (-13) or = -13 + 7n or = 7n - 13 *7n + (-13) or 7n -13 is the simplified form because there are no parentheses and no like terms.

Simplify: 3x + 0.5x + 4 + 0.8

3x + 0.5x + 4 + 0.8 Simplify the expression: STEP 1: There are no parentheses, so go to STEP 2. STEP 2: Combine the like terms: Add 3x and 0.5x to get 3.5x. Add 4 and 0.8 to get 4.8. 3x + 0.5x + 4 + 0.8 = (3x + 0.5x) + (4 + 0.8) = 3.5x + 4.8 *3.5x + 4.8 is the simplified form because there are no parentheses and no like terms.

Simplify using the Distributive Property: 3x(2x - 6)

3x(2x - 6) = (3x ⋅ 2x) - (3x ⋅ 6) = 6x² - 18x or = 6x² + (-18x) or = -18x + 6x²

Simplify using the Distributive Property: 4(3 + x + x + y)

4(3 + x + x + y) = (4 ⋅ 3) + (4 ⋅ x) + (4 ⋅ x) + (4 ⋅ y) = 12 + 4x + 4x + 4y = 12 + 8x + 4y or = 8x + 4y + 12 *Or, you can simplify the initial expression and only distribute to 3 terms instead of 2. 4(3 + x + x + y) = 4(3 + 2x + y) = (4 ⋅ 3) + (4 ⋅ 2x) + (4 ⋅ y) = 12 + 8x + 4y or = 8x + 4y + 12

Simplify using the Distributive Property: 4(5a + 3b - 2b)

4(5a + 3b - 2b) = (4 ⋅ 5a) + (4 ⋅ 3b) - (4 ⋅ 2b) = 20a + 12b - 8b = 20a + [12b - 8b] = 20a + 4b

Simplify: 6(x - 1) + 5(x + 4)

6(x - 1) + 5(x + 4) Simplify the expression: STEP 1: Remove parentheses from left to right using the Distributive Property. 6(x - 1) + 5(x + 4) = (6 ⋅ x) - (6 ⋅ 1) + (5 ⋅ x) + (5 ⋅ 4) = 6x - 6 + 5x + 20 = 6x + (-6) + 5x + 20 = 6x + 5x + (-6) + 20 STEP 2: Combine like terms. Add 6x and 5x to get 11x. Subtract the constants to get 14. = 6x + 5x + (-6) + 20 = (6x + 5x) + (-6 + 20) = 11x + 14 *11x + 14 is the simplified form because there are no parentheses and no like terms.

Simplify: 8 - (2x + 3)

8 - (2x + 3) Simplify the expression: STEP 1: Remove parentheses from left to right using the Distributive Property. This is a tricky one! There is a hidden -1 in front of the parentheses (because of the subtraction sign) that you need to distribute. When you multiply a number or expression by -1, it will turn the number or expression into its opposite. 8 - (2x + 3) = 8 + (-1)(2x + 3) = 8 + (-1 ⋅ 2x) + (-1 ⋅ 3) = 8 + (-2x) + (-3) = -2x + 8 + (-3) STEP 2: Combine like terms. Combine 8 and -3 to get 5. -2x + 8 + (-3) = -2x + [8 + (-3)] = -2x + 5 *-2x + 5 is the simplified form because there are no parentheses and no like terms.

What are the like terms in the algebraic expression (polynomial) 4x + 5.5y + 6y - 7 − x + 3?

4x, -1x 5.5y, 6y 3,-7

Simplify using the Distributive Property: (2x + 3)6

(2x + 3)6 *When the number to be distributed comes after the parentheses, still ALWAYS distribute to the first term in the parentheses (from left to right). (2x + 3)6 or 6(2x + 3) = (6 ⋅ 2x) + (6 ⋅ 3) = 12x + 18

Simplify using the Distributive Property: 2(3 - 1)

2(3 - 1) = (2 × 3) - (2 × 1) = 6 - 2 = 4

Simplify: 4(6 + 8) + 2(9 - 7)

4(6 + 8) + 2(9 - 7) Use PEMDAS and The Distributive Property to simplify from left to right. 4(6 + 8) + 2(9 - 7) = [(4 × 6) + (4 × 8)] + [(2 × 9) - (2 × 7)] = (24 + 32) + (18 - 14) or = 24 + 32 + 18 + (-14) = 56 + 4 = 60

What are the coefficient(s) in the algebraic expression (polynomial) 4x + 5.5y + 6y - 7 − x + 3?

4, 5.5, 6, and -1

Simplify using the Distributive Property: (y - 2)2y

(y - 2)2y *When the expression to be distributed comes after the parentheses, still ALWAYS distribute to the first term in the parentheses (from left to right). (y - 2)2y or 2y(y - 2) = (2y ⋅ y) - (2y ⋅ 2) = 2y² - 4y or = 2y² + (-4y) or = -4y + 2y²

Simplify using the Distributive Property: 2[2y + (-5)]

2[2y + (-5)] = (2 ⋅ 2y) + (2 ⋅ -5) = 4y + (-10) or = -10 + 4y or = 4y − 10

How many terms are in the algebraic expression (polynomial) 4x + 5.5y + 6y - 7 − x + 3?

There are 6 terms in the (polynomial) 4x + 5.5y + 6y - 7 − x + 3. They are 4x, 5.5y, 6y, -7, -1x, and 3.