ASVAB Study Guide
Pre-Algebra · Lesson 1 of 3

Combining Like Terms and Adding and Subtracting Polynomials

Key rules and formulas, worked examples and 9 practice problems with step-by-step solutions.

1. Combining Like Terms

Grouping terms with the same variable and exponent

$$ax^n + bx^n = (a+b)x^n$$

Only add or subtract the coefficients (numbers). Variables and exponents stay the same.

Example

Simplify $$5x^2 + 3x - 2x^2 + 7x - 4$$

Show solution

Step 1: Group like terms:

  • $$x^2$$ terms: $$(5x^2 - 2x^2) = (5 - 2)x^2 = 3x^2$$
  • x terms: $$(3x + 7x) = (3 + 7)x = 10x$$
  • Constant: -4

Step 2: Combine all simplified terms.

Result: $$3x^2 + 10x - 4$$

Practice problems

  1. Problem 1

    Simplify $$4x + 7x - 3x$$

    Show solution

    Answer: $$8x$$

    Step 1: Identify that all terms are like terms (variable x).

    Step 2: Combine coefficients: $$(4 + 7 - 3)x = 8x$$.

    => Correct Answer: A ($$8x$$)

  2. Problem 2

    Simplify $$3x^2 + 5x + 2x^2 - 8x + 6$$

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    Answer: $$5x^2 - 3x + 6$$

    Step 1: Group $$x^2$$ terms: $$(3x^2 + 2x^2) = 5x^2$$.

    Step 2: Group x terms: $$(5x - 8x) = -3x$$.

    Step 3: Keep constant term: + 6.

    => Combined expression $$= 5x^2 - 3x + 6$$.

    => Correct Answer: A ($$5x^2 - 3x + 6$$)

  3. Problem 3

    Simplify $$4x^2 y - 2xy^2 + 3x^2 y + 5xy^2 - 7$$

    Show solution

    Answer: $$7x^2 y + 3xy^2 - 7$$

    Step 1: Group $$x^2 y$$ terms: $$(4x^2 y + 3x^2 y) = 7x^2 y$$.

    Step 2: Group $$xy^2$$ terms: $$(-2xy^2 + 5xy^2) = 3xy^2$$.

    Step 3: Keep constant: -7.

    => Combined expression $$= 7x^2 y + 3xy^2 - 7$$.

    => Correct Answer: A ($$7x^2 y + 3xy^2 - 7$$).

2. Adding Polynomials

Adding two or more expressions together

(Poly A) + (Poly B) = Combine all like terms

Remove parentheses and combine all like terms. Signs stay the same.

Example

Add $$(3x^2 - 4x + 5) + (2x^2 + 6x - 1)$$

Show solution

Step 1: Remove parentheses: $$3x^2 - 4x + 5 + 2x^2 + 6x - 1$$.

Step 2: Group like terms:

  • $$(3x^2 + 2x^2) = 5x^2$$
  • $$(-4x + 6x) = 2x$$
  • $$(5 - 1) = 4$$

Result: $$5x^2 + 2x + 4$$

Practice problems

  1. Problem 4

    Add $$(2x + 5) + (3x - 2)$$

    Show solution

    Answer: $$5x + 3$$

    Step 1: Remove parentheses: $$2x + 5 + 3x - 2$$.

    Step 2: Combine x terms: $$2x + 3x = 5x$$.

    Step 3: Combine constants: $$5 - 2 = 3$$.

    => Result $$= 5x + 3$$.

    => Correct Answer: A ($$5x + 3$$)

  2. Problem 5

    Add $$(4x^2 - 3x + 7) + (x^2 + 5x - 10)$$

    Show solution

    Answer: $$5x^2 + 2x - 3$$

    Step 1: Remove parentheses: $$4x^2 - 3x + 7 + x^2 + 5x - 10$$.

    Step 2: Combine $$x^2$$ terms: $$4x^2 + x^2 = 5x^2$$.

    Step 3: Combine x terms: $$-3x + 5x = 2x$$.

    Step 4: Combine constants: $$7 - 10 = -3$$.

    => Result $$= 5x^2 + 2x - 3$$.

    => Correct Answer: A ($$5x^2 + 2x - 3$$)

  3. Problem 6

    Add $$(2x^3 - 5x^2 + 4) + (3x^3 + 2x^2 - 6x - 1)$$

    Show solution

    Answer: $$5x^3 - 3x^2 - 6x + 3$$

    Step 1: Remove parentheses: $$2x^3 - 5x^2 + 4 + 3x^3 + 2x^2 - 6x - 1$$.

    Step 2: Combine $$x^3$$ terms: $$2x^3 + 3x^3 = 5x^3$$.

    Step 3: Combine $$x^2$$ terms: $$-5x^2 + 2x^2 = -3x^2$$.

    Step 4: Keep x term: $$-6x$$.

    Step 5: Combine constants: $$4 - 1 = 3$$.

    => Result $$= 5x^3 - 3x^2 - 6x + 3$$.

    => Correct Answer: A ($$5x^3 - 3x^2 - 6x + 3$$)

3. Subtracting Polynomials

Subtracting one expression from another

(Poly A) - (Poly B) = Poly A + (-Poly B)

Distribute the negative sign: Flip the signs of EVERY term inside the second parenthesis, then combine like terms.

Example

Subtract $$(5x^2 + 3x - 2) - (2x^2 - 4x + 1)$$

Show solution

Step 1: Distribute negative sign to second polynomial:

  • $$-(2x^2 - 4x + 1) = -2x^2 + 4x - 1$$.

Step 2: Rewrite expression: $$5x^2 + 3x - 2 - 2x^2 + 4x - 1$$.

Step 3: Group and combine like terms:

  • $$(5x^2 - 2x^2) = 3x^2$$
  • $$(3x + 4x) = 7x$$
  • $$(-2 - 1) = -3$$

Result: $$3x^2 + 7x - 3$$

Practice problems

  1. Problem 7

    Subtract $$(6x + 8) - (2x + 3)$$

    Show solution

    Answer: $$4x + 5$$

    Step 1: Distribute minus sign: $$6x + 8 - 2x - 3$$.

    Step 2: Combine x terms: $$6x - 2x = 4x$$.

    Step 3: Combine constants: $$8 - 3 = 5$$.

    => Result $$= 4x + 5$$.

    => Correct Answer: A ($$4x + 5$$)

  2. Problem 8

    Subtract $$(7x^2 - 4x + 2) - (3x^2 + 2x - 5)$$

    Show solution

    Answer: $$4x^2 - 6x + 7$$

    Step 1: Distribute minus sign to second set of parentheses:

    • $$-(3x^2 + 2x - 5) = -3x^2 - 2x + 5$$.

    Step 2: Rewrite expression: $$7x^2 - 4x + 2 - 3x^2 - 2x + 5$$.

    Step 3: Combine $$x^2$$ terms: $$7x^2 - 3x^2 = 4x^2$$.

    Step 4: Combine x terms: $$-4x - 2x = -6x$$.

    Step 5: Combine constants: $$2 + 5 = 7$$.

    => Result $$= 4x^2 - 6x + 7$$.

    => Correct Answer: A ($$4x^2 - 6x + 7$$)

  3. Problem 9

    Subtract $$(4x^3 - x^2 + 3x) - (x^3 - 5x^2 - 2x + 8)$$

    Show solution

    Answer: $$3x^3 + 4x^2 + 5x - 8$$

    Step 1: Distribute negative sign:

    • $$-(x^3 - 5x^2 - 2x + 8) = -x^3 + 5x^2 + 2x - 8$$.

    Step 2: Rewrite expression: $$4x^3 - x^2 + 3x - x^3 + 5x^2 + 2x - 8$$.

    Step 3: Combine $$x^3$$ terms: $$4x^3 - x^3 = 3x^3$$.

    Step 4: Combine $$x^2$$ terms: $$-x^2 + 5x^2 = 4x^2$$.

    Step 5: Combine x terms: $$3x + 2x = 5x$$.

    Step 6: Keep constant: -8.

    => Result $$= 3x^3 + 4x^2 + 5x - 8$$.

    => Correct Answer: A ($$3x^3 + 4x^2 + 5x - 8$$)