Polynomials
Expand, factor, and use the remainder theorem.
Pen and paper is fine · no calculator needed why?
Opens at level 26. You're level 1. You can read and practice here now.
the idea
A polynomial adds up whole-number powers of x, each times a number, like x² − 7x + 12. The number in front of a power is its coefficient, and the plain number is the constant term.
Expanding multiplies brackets out. Factoring runs it backward, turning a sum into brackets. The remainder theorem saves a long division: when p(x) is divided by x − r, the remainder is p(r), the value you get by putting r in for x.
techniques
First, outer, inner, last
- First terms multiply to give the x² term.
- Outer plus inner give the x term: add those two products.
- Last terms multiply to give the constant.
- Carry each sign with its number.
worked example
Expand (2x + 3)(x − 5). What is the coefficient of x?
- Outer: 2x × (−5) = −10x.
- Inner: 3 × x = 3x.
- Add them: −10 + 3 = −7.
- Full expansion: 2x² − 7x − 15.
Answer: −7
Product and sum
- a and b multiply to the constant p, and add to s, the number in front of −x.
- List the factor pairs of p and pick the pair with that sum.
- Check by multiplying the brackets back out.
worked example
x² − 9x + 20 = (x − a)(x − b) with a < b. What is b?
- Pairs that multiply to 20: 1 and 20, 2 and 10, 4 and 5.
- Only 4 and 5 add to 9.
- So a = 4 and b = 5.
Answer: 5
Put r in for x
- Dividing by x − 2 means r is 2. Dividing by x + 2 means r is −2.
- Put r in for x everywhere. Bracket a negative r before you raise it to a power.
- Work it out. That value is the remainder.
worked example
What is the remainder when x³ + 2x² − 5 is divided by x + 1?
- Dividing by x + 1 means putting in −1.
- (−1)³ + 2 × (−1)² − 5 = −1 + 2 − 5 = −4.
Answer: −4
watch out for
- Giving the constant term when the question asks for the coefficient of x. The x term comes from outer plus inner.
- Picking a pair that multiplies to the constant but does not add to the middle number.
- Putting in 2 when dividing by x + 2. The sign flips: x + 2 means putting in −2.
practice
Expand brackets
worked example
Expand (x + 1)(x − 7). What is the coefficient of x?
Answer: -6
- Outer: x × (−7) = −7x. Inner: 1 × x = x.
- Add them: the coefficient of x is −7 + 1 = −6.
- Full expansion: x² − 6x − 7.
Factor a trinomial
worked example
x² − 17x + 70 = (x − a)(x − b) with a < b. Enter a, b.
Answer: (7, 10)
- Find two numbers that multiply to 70 and add to 17: 7 and 10.
- So x² − 17x + 70 = (x − 7)(x − 10).
Remainder theorem
worked example
What is the remainder when 2x³ + 3x² + 3x + 2 is divided by x − 1?
Answer: 10
- Call the polynomial p(x). By the remainder theorem, the remainder is p(1).
- p(1) = 2 × 1³ + 3 × 1² + 3 × 1 + 2 = 2 + 3 + 3 + 2 = 10.