GED Math Expressions and Polynomials Summary

1. Group the like terms: (4x − 2x) + (7 + 3).
2. 4x − 2x = 2x.
3. 7 + 3 = 10.
4. 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.

1. The $65 fee is charged once, so it stays as 65.
2. The $40 is charged for each hour, so it is 40 × h = 40h.
3. 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.

1. Substitute: 3(4)² − 2(5).
2. Exponent first: 4² = 16.
3. Multiply: 3 × 16 = 48 and 2 × 5 = 10.
4. 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.

1. Multiply −3 by each term inside the parentheses.
2. −3 × 2x = −6x.
3. −3 × (−5) = +15.
4. Result: −6x + 15.

Check: Try x = 1. −3(2 − 5) = −3(−3) = 9, and −6(1) + 15 = 9.

Wrong choices

• −6x − 15: −6x − 15 forgets that a negative times a negative is positive.
• −6x − 5: −6x − 5 multiplies only the first term by −3.
• 6x + 15: 6x + 15 drops the negative sign on the first term.

1. First: x × x = x².
2. Outer: x × (−3) = −3x.
3. Inner: 4 × x = 4x.
4. Last: 4 × (−3) = −12.
5. Combine: x² − 3x + 4x − 12 = x² + x − 12.

Check: Try x = 2. (2 + 4)(2 − 3) = 6 × (−1) = −6, and 4 + 2 − 12 = −6.

Wrong choices

• x² − 12: x² − 12 multiplies only the first terms and the last terms; the middle terms are missing.
• x² + 7x − 12: x² + 7x − 12 uses +3 instead of −3 for the middle term (4 + 3 = 7).
• x² − x − 12: x² − x − 12 has the middle sign backward: −3x + 4x is +x, not −x.

1. Find two numbers that multiply to −14 and add to −5.
2. −7 × 2 = −14 and −7 + 2 = −5.
3. So x² − 5x − 14 = (x − 7)(x + 2).

Check: (x − 7)(x + 2) = x² + 2x − 7x − 14 = x² − 5x − 14.

Wrong choices

• (x + 7)(x − 2): (x + 7)(x − 2) gives a middle term of +5x.
• (x − 7)(x − 2): (x − 7)(x − 2) gives +14 at the end and −9x in the middle.
• (x − 14)(x + 1): (x − 14)(x + 1) gives a middle term of −13x.

1. Change the sign of each term in the second polynomial: −2x² + 4x − 6.
2. Add: (5x² − 2x²) + (3x + 4x) + (−2 − 6).
3. Result: 3x² + 7x − 8.

Check: Try x = 1. First: 5 + 3 − 2 = 6. Second: 2 − 4 + 6 = 4. 6 − 4 = 2, and 3 + 7 − 8 = 2.

Wrong choices

• 3x² − x + 4: 3x² − x + 4 changes the sign of only the first term of the second polynomial.
• 3x² + 7x + 4: 3x² + 7x + 4 forgets to change the sign of the +6.
• 3x² − x − 8: 3x² − x − 8 forgets to change the sign of the −4x.

1. The GCF of 6 and 9 is 3.
2. The GCF of x³ and x² is x².
3. Take out 3x²: 6x³ ÷ 3x² = 2x and 9x² ÷ 3x² = 3.
4. Result: 3x²(2x − 3).

Check: Multiply back: 3x² × 2x = 6x³ and 3x² × 3 = 9x².

Wrong choices

• 3x(2x − 3): 3x(2x − 3) multiplies back to 6x² − 9x; the powers of x are too low.
• 3x²(2x − 9): 3x²(2x − 9) multiplies back to 6x³ − 27x²; the 9 was not divided by 3.
• 3x³(2 − 3x): 3x³(2 − 3x) multiplies back to 6x³ − 9x⁴; the second term has the wrong power.

1. Factor the top as a difference of squares: x² − 9 = (x − 3)(x + 3).
2. Cancel the common factor (x + 3).
3. Result: x − 3.

Check: Try x = 5. (25 − 9) ÷ (5 + 3) = 16 ÷ 8 = 2, and 5 − 3 = 2.

Wrong choices

• x + 3: x + 3 is the factor that cancels out, not what is left.
• x² − 3: x² − 3 divides only the 9 by 3; you cannot cancel parts of a sum.
• x − 9: x − 9 cancels x² with x, which is not allowed in a sum.

1. Length = 2w + 3.
2. Area = width × length = w(2w + 3).
3. Distribute: w × 2w + w × 3 = 2w² + 3w.

Check: If w = 2, the length is 7 and the area is 14. 2(2)² + 3(2) = 8 + 6 = 14.

Wrong choices

• 2w + 3: 2w + 3 is the length, not the area.
• 6w + 6: 6w + 6 is the perimeter: 2w + 2(2w + 3).
• 2w² + 3: 2w² + 3 multiplies only the first term by w; w must multiply the 3 as well.