GED Math: Expressions and Polynomials Concepts
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Overview
This chapter covers algebraic expressions: writing them from word descriptions, evaluating them by substituting numbers, combining like terms, and adding, subtracting, multiplying, and factoring polynomials. It also covers simple rational expressions, which are fractions with variables in them.
Like terms have the same variable raised to the same power, and only like terms can be combined. The distributive property multiplies the number outside the parentheses by every term inside, and the sign goes along. When one polynomial is subtracted from another, the sign of every term in the second polynomial changes.
To multiply two binomials, multiply each term in the first by each term in the second, often remembered as FOIL (first, outer, inner, last), and then combine like terms. Factoring reverses multiplication: take out the greatest common factor first, then look for two numbers that multiply to the constant term and add to the middle coefficient.
A rational expression is simplified by factoring the top and bottom and canceling a common factor. Only whole factors cancel, never single terms in a sum. To add rational expressions, rewrite them with a common denominator first.
Key concepts
Like terms
Terms with the same variable and exponent, such as 4x and −2x. Add or subtract their numbers and keep the variable part.
Memory tipx and x² are not like terms.
Distributive property
a(b + c) = ab + ac. Multiply the outside number by every term inside, keeping track of signs.
Memory tip−3(2x − 5) = −6x + 15, because negative × negative is positive.
Multiplying binomials (FOIL)
Multiply First, Outer, Inner, and Last terms, then combine like terms. (a − b)² = a² − 2ab + b².
Memory tipA squared binomial always has a middle term.
Factoring
Write an expression as a product. Take out the GCF first, then factor trinomials like x² + bx + c into (x + m)(x + n), where m × n = c and m + n = b. A difference of squares a² − b² = (a − b)(a + b).
Memory tipMultiply your factors back to check.
Rational expressions
Fractions with variables. Simplify by factoring and canceling common factors. Add by using a common denominator.
Memory tipTest your simplified answer with a small number such as x = 2.
Often tested
- When subtracting polynomials, change the sign of every term in the second one.
- Evaluate with parentheses around substituted negative numbers.
- A fixed fee is added once; a per-unit charge is multiplied by the variable.
- You cannot cancel terms that are added or subtracted, only factors.
3 sample questions
Verified practice questions from this unit.
Question 1. Simplify: 4x + 7 − 2x + 3
- ① 2x + 10
- ② 12x
- ③ 6x + 10
- ④ 2x + 4
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Answer: ① 2x + 10
Key point: Combine x-terms with x-terms, numbers with numbers
- Group the like terms: (4x − 2x) + (7 + 3).
- 4x − 2x = 2x.
- 7 + 3 = 10.
- The result is 2x + 10.
Check: Try x = 1. The original is 4 + 7 − 2 + 3 = 12, and 2(1) + 10 = 12.
Wrong choices
- 6x + 10: 6x + 10 adds 4x and 2x, but the 2x is being subtracted.
- 12x: 12x combines numbers and x-terms together; only like terms can be combined.
- 2x + 4: 2x + 4 subtracts 3 from 7, but the 3 is being added.
Question 2. A plumber charges a $65 visit fee plus $40 for each hour of work. Which expression gives the total charge for h hours?
- ① 40 + 65h
- ② 65h + 40h
- ③ 105h
- ④ 65 + 40h
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Answer: ④ 65 + 40h
Key point: Fixed fee + (rate × number of hours)
- The $65 fee is charged once, so it stays as 65.
- The $40 is charged for each hour, so it is 40 × h = 40h.
- Total: 65 + 40h.
Check: For 2 hours, 65 + 40(2) = $145, which is $65 + $40 + $40.
Wrong choices
- 40 + 65h: 40 + 65h charges $65 for each hour and $40 once, which is backward.
- 105h: 105h charges both amounts every hour, but the visit fee is charged only once.
- 65h + 40h: 65h + 40h also charges the visit fee every hour.
Question 3. What is the value of 3a² − 2b when a = 4 and b = 5?
- ① 38
- ② 58
- ③ 134
- ④ 14
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Answer: ① 38
Key point: Substitute, then square before you multiply
- Substitute: 3(4)² − 2(5).
- Exponent first: 4² = 16.
- Multiply: 3 × 16 = 48 and 2 × 5 = 10.
- Subtract: 48 − 10 = 38.
Check: 3 × 4 × 4 = 48, and 48 − 10 = 38.
Wrong choices
- 134: 134 squares 3 × 4 = 12 to get 144; only a is squared, not 3a.
- 14: 14 treats a² as 2 × a (8), so 3 × 8 − 10 = 14.
- 58: 58 adds 2b instead of subtracting it.
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