A2x² − 2x − 12 — only the first term of the second polynomial is negated
B2x² + 6x − 12 — the negative is distributed to every term of the second polynomial
C4x² − 2x + 2 — the polynomials are added instead of subtracted
D2x² − 6x + 2 — the sign of the x-term is negated twice
The key step: distribute the negative to every term of (x² − 4x + 7), giving (−x² + 4x − 7). Note that −(−4x) = +4x, which is the step most commonly missed. Then combine like terms column by column: (3x² − x²) = 2x², (2x + 4x) = 6x, (−5 − 7) = −12. Result: 2x² + 6x − 12. Answer A is the classic error — only negating the first term (getting −x² but leaving −4x unchanged), which gives 2x² − 2x − 12.
Question 2 Multiple Choice
Which of the following is a valid step when adding 5x³ + 2x to 3x² + x?
ACombine 5x³ and 3x² to get 8x⁵, since both terms contain x
BCombine 2x and x to get 3x, since both terms are x-terms with the same exponent
CCombine 5x³ and x to get 5x⁴, multiplying the exponents
DCombine all four terms into one expression since they all involve x
Only like terms — same variable AND same exponent — can be combined. 2x and x both have variable x to the first power, so they combine: 2x + x = 3x. The terms 5x³ and 3x² cannot be combined because their exponents differ (3 vs. 2): they belong to different 'species.' Exponents are labels that identify the species, not numbers to add. The result is 5x³ + 3x² + 3x — four terms become three because only one pair of like terms existed.
Question 3 True / False
When subtracting one polynomial from another, you mainly need to change the sign of the first term of the polynomial being subtracted.
TTrue
FFalse
Answer: False
This is the most common error in polynomial subtraction. The subtraction sign applies to the entire polynomial — every term within it — because the expression A − (B + C + D) must be rewritten as A + (−B) + (−C) + (−D). Changing only the first term's sign leaves the remaining terms with incorrect signs, producing a wrong answer. Every term inside the parentheses must be negated before combining like terms.
Question 4 True / False
When adding or subtracting polynomials, the exponents of like terms are added together along with their coefficients.
TTrue
FFalse
Answer: False
Exponents are never added when combining like terms — they remain unchanged. When you combine 3x² + x², the exponents stay at 2: the result is 4x², not 4x⁴. Exponents serve as labels identifying which 'species' of term you are dealing with (the x²-species, the x-species, the constant-species). Only coefficients are added. Exponent addition belongs to multiplication, not addition — this is a critical distinction to keep clear.
Question 5 Short Answer
Why must the negative sign be distributed to every term when subtracting a polynomial, rather than just the first term?
Think about your answer, then reveal below.
Model answer: Because subtraction means adding the opposite of the entire polynomial, not just its first term. The expression A − (B + C) is equivalent to A + (−1)(B + C) = A − B − C. The negative one multiplies every term inside the parentheses by the distributive property. Negating only the first term treats the parentheses as if they weren't there, which changes the mathematical meaning of the expression.
The parentheses in polynomial subtraction are not decorative — they indicate that the negative sign applies to the whole group. This is the same distributive property students use when expanding −2(x + 3) = −2x − 6. Polynomial subtraction is just a special case where the coefficient is −1. Keeping this principle clear prevents the most common error in this topic.