Question

1. draw algebra tiles to model the expression, combine like terms, and write simplified expression.

a) $4x + 1 + x - 5$
b) $-2 + 3x + 5x + 4x + 1$
c) $2 + x^2 + 3x + 2x^2 - 3$
d) $2x + 3x^2 + 3x - 2x^2 + 4$

1. write the simplified expression by combining like terms. use algebra tiles if needed.

a) $3x - 2 - 2x + 4$
b) $x + 3 - 2x - 2 - x$
c) $1 - x^2 + 2x + 2x^2 - 2$
d) $x^2 - 2x - 2x^3 + x^2 - 3 + 3x + 1$
e) $3x^2 + 3 - x + x^2 - 6 + x + 7$

bonus extra credit: $4x^3 + x + 2x^2 + 2x^3 - x^2 + 3x - 8 + 2$

Explanation:

Let's solve part 4a first: \( 4x + 1 + x - 5 \)

Step 1: Identify like terms

Like terms are terms with the same variable (or constant terms). Here, the \( x \)-terms are \( 4x \) and \( x \), and the constant terms are \( 1 \) and \( -5 \).

Step 2: Combine \( x \)-terms

\( 4x + x = 5x \)

Step 3: Combine constant terms

\( 1 - 5 = -4 \)

Step 4: Write the simplified expression

Combine the results from Step 2 and Step 3: \( 5x - 4 \)

Now part 4b: \( -2 + 3x + 5x + 1 \)

Step 1: Identify like terms

\( x \)-terms: \( 3x \) and \( 5x \); constant terms: \( -2 \) and \( 1 \)

Step 2: Combine \( x \)-terms

\( 3x + 5x = 8x \)

Step 3: Combine constant terms

\( -2 + 1 = -1 \)

Step 4: Write the simplified expression

\( 8x - 1 \)

Part 4c: \( 2 + x^2 + 3x + 2x^2 - 3 \)

Step 1: Identify like terms

\( x^2 \)-terms: \( x^2 \) and \( 2x^2 \); \( x \)-term: \( 3x \); constant terms: \( 2 \) and \( -3 \)

Step 2: Combine \( x^2 \)-terms

\( x^2 + 2x^2 = 3x^2 \)

Step 3: Combine constant terms

\( 2 - 3 = -1 \)

Step 4: Write the simplified expression

\( 3x^2 + 3x - 1 \)

Part 4d: \( 2x + 3x^2 + 3x - 2x^2 + 4 \)

Step 1: Identify like terms

\( x^2 \)-terms: \( 3x^2 \) and \( -2x^2 \); \( x \)-terms: \( 2x \) and \( 3x \); constant term: \( 4 \)

Step 2: Combine \( x^2 \)-terms

\( 3x^2 - 2x^2 = x^2 \)

Step 3: Combine \( x \)-terms

\( 2x + 3x = 5x \)

Step 4: Write the simplified expression

\( x^2 + 5x + 4 \)

Part 5a: \( 3x - 2 - 2x + 4 \)

Step 1: Identify like terms

\( x \)-terms: \( 3x \) and \( -2x \); constant terms: \( -2 \) and \( 4 \)

Step 2: Combine \( x \)-terms

\( 3x - 2x = x \)

Step 3: Combine constant terms

\( -2 + 4 = 2 \)

Step 4: Write the simplified expression

\( x + 2 \)

Part 5b: \( x + 3 - 2x - 2 - x \)

Step 1: Identify like terms

\( x \)-terms: \( x \), \( -2x \), and \( -x \); constant terms: \( 3 \) and \( -2 \)

Step 2: Combine \( x \)-terms

\( x - 2x - x = -2x \)

Step 3: Combine constant terms

\( 3 - 2 = 1 \)

Step 4: Write the simplified expression

\( -2x + 1 \)

Part 5c: \( 1 - x^2 + 2x + 2x^2 - 2 \)

Step 1: Identify like terms

\( x^2 \)-terms: \( -x^2 \) and \( 2x^2 \); \( x \)-term: \( 2x \); constant terms: \( 1 \) and \( -2 \)

Step 2: Combine \( x^2 \)-terms

\( -x^2 + 2x^2 = x^2 \)

Step 3: Combine constant terms

\( 1 - 2 = -1 \)

Step 4: Write the simplified expression

\( x^2 + 2x - 1 \)

Part 5d: \( x^2 - 2x - 2x^3 + x^2 - 3 + 3x + 1 \)

Step 1: Identify like terms

\( x^3 \)-term: \( -2x^3 \); \( x^2 \)-terms: \( x^2 \) and \( x^2 \); \( x \)-terms: \( -2x \) and \( 3x \); constant terms: \( -3 \) and \( 1 \)

Step 2: Combine \( x^3 \)-term

\( -2x^3 \) (only one \( x^3 \)-term)

Step 3: Combine \( x^2 \)-terms

\( x^2 + x^2 = 2x^2 \)

Step 4: Combine \( x \)-terms

\( -2x + 3x = x \)

Step 5: Combine constant terms

\( -3 + 1 = -2 \)

Step 6: Write the simplified expression

\( -2x^3 + 2x^2 + x - 2 \)

Part 5e: \( 3x^2 + 3 - x + x^2 - 6 + x + 7 \)

Step 1: Identify like terms

\( x^2 \)-terms: \( 3x^2 \) and \( x^2 \); \( x \)-terms: \( -x \) and \( x \); constant terms: \( 3 \), \( -6 \), and \( 7 \)

Step 2: Combine \( x^2 \)-terms

\( 3x^2 + x^2 = 4x^2 \)

Step 3: Combine \( x \)-terms

\( -x + x = 0 \) (they cancel out)

Step 4: Combine constant terms

\( 3 - 6 + 7 = 4 \)

Step 5: Write the simplified expression

\( 4x^2 + 4 \)

BONUS: \( 4x^3 + x + 2x^2 + 2x^3 - x^2 + 3x - 8 + 2 \)

Step 1: Identify like terms

\( x^3 \)-terms: \( 4x^3 \) and \( 2x^3 \); \( x^2 \)-terms: \( 2x^2 \) and \( -x^2 \); \( x \)-terms: \( x \) and \( 3x \); constant terms: \( -8 \) and \( 2 \)

St…

Answer:

s:
4a: \( \boldsymbol{5x - 4} \)
4b: \( \boldsymbol{8x - 1} \)
4c: \( \boldsymbol{3x^2 + 3x - 1} \)
4d: \( \boldsymbol{x^2 + 5x + 4} \)
5a: \( \boldsymbol{x + 2} \)
5b: \( \boldsymbol{-2x + 1} \)
5c: \( \boldsymbol{x^2 + 2x - 1} \)
5d: \( \boldsymbol{-2x^3 + 2x^2 + x - 2} \)
5e: \( \boldsymbol{4x^2 + 4} \)
BONUS: \( \boldsymbol{6x^3 + x^2 + 4x - 6} \)