courses › Algebra

Polynomials

level 26 course

Expand, factor, and use the remainder theorem.

Pen and paper is fine · no calculator needed why?

Opens at level 26.

Builds on: Linear equations (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

Multiplying two brackets, like (2x + 3)(x − 5).

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

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

Factoring x² − sx + p into (x − a)(x − b).

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

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

The remainder when a polynomial is divided by x − r or x + r.

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

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

Tips by skill

TipExpand brackets: First terms give x², outer plus inner give x, last terms give the constant. Keep each sign with its number.
TipFactor a trinomial: In (x − a)(x − b), a and b multiply to the constant and add to the number in front of −x.
TipRemainder theorem: Divided by x − r? Put r in for x. Divided by x + r? Put in −r. The value you get is the remainder.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Expand brackets par 16 s not tried yet
worked example
Expand (x + 1)(x − 7). What is the coefficient of x?

Answer: -6
1. Outer: x × (−7) = −7x. Inner: 1 × x = x.
2. Add them: the coefficient of x is −7 + 1 = −6.
3. Full expansion: x² − 6x − 7.
Factor a trinomial par 20 s not tried yet
worked example
x² − 17x + 70 = (x − a)(x − b) with a < b. Enter a, b.

Answer: (7, 10)
1. Find two numbers that multiply to 70 and add to 17: 7 and 10.
2. So x² − 17x + 70 = (x − 7)(x − 10).
Remainder theorem par 18 s not tried yet
worked example
What is the remainder when 2x³ + 3x² + 3x + 2 is divided by x − 1?

Answer: 10
1. Call the polynomial p(x). By the remainder theorem, the remainder is p(1).
2. p(1) = 2 × 1³ + 3 × 1² + 3 × 1 + 2 = 2 + 3 + 3 + 2 = 10.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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