Tayarri LCAT
LCAT Math, Advanced Math

Polynomials for the LCAT: lesson and practice

To subtract polynomials, change the sign of every term in the second bracket. To multiply, multiply every term by every term. The zeros of a polynomial come from its factors: (x − 3) gives the zero x = 3. The remainder when p(x) is divided by (x − a) is p(a).

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No calculator on the LCAT. LUMS’s official guidelines say calculators are not allowed, so every practice question here can be solved by hand.
TaskExample
Subtract(3x² + 2x) − (x² − 5x) = 2x² + 7x
Multiply(x + 3)(x − 5) = x² − 2x − 15
Zerosp(x) = 2(x − 4)(x + 1) → 4 and −1
Remainderp(x) ÷ (x − 2) leaves p(2)
Simplify(x² − 9)/(x − 3) = x + 3
Check any "equivalent expression" answer by substituting x = 2 into both expressions.

Common mistakes

Not distributing the minus sign−(x² − 5x) = −x² + 5x.
Missing the middle term(x + 3)(x − 5) has an x-term: −5x + 3x.
Cancelling terms, not factorsFactor first, then cancel whole factors only.

Sample questions with answers

1. Which is equivalent to (3x² − 5x) − (2x² − x)?
  1. x² − 6x
  2. x² − 4x (correct)
  3. 5x² − 6x
  4. x² + 4x

−(2x² − x) = −2x² + x, so the result is x² − 4x.

2. Which is a zero of p(x) = 2(x − 2)(x + 1)²?
  1. 1
  2. −1 (correct)
  3. 2x
  4. −2

x + 1 = 0 when x = −1 (2 is also a zero).

3. Which is equivalent to (x² − 6x + 8)/(x − 4)?
  1. x − 6
  2. x + 2
  3. x − 2 (correct)
  4. x − 4

x² − 6x + 8 = (x − 4)(x − 2).

Frequently asked questions

What is the remainder theorem?

The remainder when p(x) is divided by (x − a) equals p(a).

How do I simplify rational expressions?

Factor the numerator and denominator completely, then cancel matching factors.

Is polynomials tested on the LCAT?

Yes. It falls under Advanced Math, one of the content domains LUMS lists for the LCAT in its official sample questions.

Can I practice this without a calculator?

Yes. Every practice question here can be solved by hand, because calculators are not allowed on the LCAT.

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