Operations with polynomials

How the SAT tests this, with worked examples — plus the 1 skills it covers. Free to read, no account needed.

Adding, subtracting and multiplying polynomials follows the ordinary rules of algebra, with one idea doing most of the work: only like terms combine.
et_polynomial_ops.png
Like terms have the same variable raised to the same power. So 3x23x^2 and 5x25x^2 combine to 8x28x^2, while 3x23x^2 and 5x35x^3 cannot be combined at all.
When subtracting, the minus sign applies to every term in the bracket. Forgetting to distribute it past the first term is the most frequent error here, and the resulting wrong answer is always offered.
To multiply, every term in the first bracket multiplies every term in the second, then collect like terms. Two patterns recur often enough to recognise instantly: (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2 and (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2.
Note that (a+b)2(a+b)^2 is not a2+b2a^2 + b^2. The missing middle term 2ab2ab is one of the most reliably tested mistakes on the exam.

Worked examples

Simplify (3x2+5x)−(x2−2x)(3x^2 + 5x) - (x^2 - 2x).
Distribute the minus across both terms
3x2+5x−x2+2x=2x2+7x{3x^2 + 5x - x^2 + 2x = 2x^2 + 7x}
Note the −2x-2x became +2x+2x.
Expand (x+4)2(x+4)^2.
  • Ax2+16x^2 + 16
  • Bx2+8x+16x^2 + 8x + 16
Squaring a sum produces a middle term 2ab=8x2ab = 8x, so (B) is correct. (A) is the classic omission.

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All the skills in Operations with polynomials