Polynomial operations and factoring (Algebra II) · lesson

Adding, subtracting, and multiplying

💡 Multiply every term by every term, then combine like terms; when subtracting, change the sign of every term in the second polynomial.

Like terms have the same power of x, such as 3x² and −5x². Add or subtract their numbers.

To subtract a polynomial, change the sign of each of its terms, then combine.

To multiply, take each term of the first polynomial times each term of the second. A square like (3x + 2)² means (3x + 2)(3x + 2). Don't forget the middle term.

🧠 Let's work it out

Multiply (x − 4)(x² + 2x − 3).

Multiply x by each term: x³ + 2x² − 3x.
Multiply −4 by each term: −4x² − 8x + 12.
Combine like terms: x³ − 2x² − 11x + 12.

What do you think comes next?

polynomial: a sum of terms like 4x³ − x + 7
like terms: terms with the same variable and power
expand: multiply out so no parentheses are left

Dividing by (x − a)

💡 Synthetic division by (x − a) uses only the coefficients and the number a.

Write the coefficients in order. Use 0 for any missing power. Put a on the left. Bring down the first number.

Multiply it by a, write the result under the next number, and add. Repeat to the end. The last number is the remainder. The others are the quotient's coefficients, one power lower.

🧠 Let's work it out

Divide x³ + 2x² − 5x − 6 by (x − 2).

Use synthetic division with 2 and the coefficients 1, 2, −5, −6.
Bring down 1. Then 1 × 2 = 2, and 2 + 2 = 4.
Next, 4 × 2 = 8, and −5 + 8 = 3.
Last, 3 × 2 = 6, and −6 + 6 = 0.
The quotient is x² + 4x + 3, remainder 0.

What do you think comes next?

quotient: the answer to a division
coefficient: the number in front of a variable, like 5 in 5x²
synthetic division: a short way to divide a polynomial by (x − a)

The remainder and factor theorems

💡 The remainder from dividing p(x) by (x − a) is p(a), and (x − a) is a factor exactly when p(a) = 0.

You don't have to divide to find a remainder. Just plug a into the polynomial. Note the sign: for (x − 3) plug in 3, and for (x + 3) plug in −3.

If the remainder is 0, then (x − a) divides evenly, so it is a factor. You can use this to find an unknown number in the polynomial: set p(a) = 0 and solve.

🧠 Let's work it out

Is (x + 2) a factor of p(x) = x³ − 2x + 4?

The factor x + 2 means plug in −2.
So p(−2) = −8 + 4 + 4 = 0.
The remainder is 0, so yes, (x + 2) is a factor.

What do you think comes next?

remainder: what is left over after dividing
factor: an expression that divides another with no remainder

Factoring by grouping and trinomials

💡 Group terms to pull out a shared factor, and always check whether a factor can be factored again.

For four terms, split them into two pairs. Pull a common factor out of each pair. If the same binomial appears twice, pull it out too.

For ax² + bx + c, find two numbers that multiply to a × c and add to b. Split the middle term with them, then group.

Factor completely means keep going until no factor breaks down further. A difference of squares, like x² − 25, splits into (x − 5)(x + 5).

🧠 Let's work it out

Factor x³ − 2x² − 9x + 18 completely.

Group: x²(x − 2) − 9(x − 2).
Pull out (x − 2): (x − 2)(x² − 9).
Then x² − 9 is a difference of squares: (x − 3)(x + 3).
So the answer is (x − 2)(x − 3)(x + 3).

What do you think comes next?

factor completely: break into factors until none can be factored further
trinomial: a polynomial with three terms
difference of squares: a² − b², which factors as (a − b)(a + b)

🔗 Khan Academy: Polynomial factorization · Khan Academy (free) ↗

Sums and differences of cubes

💡 a³ − b³ = (a − b)(a² + ab + b²) and a³ + b³ = (a + b)(a² − ab + b²).

First find a and b. Ask what was cubed to make each term. For 64x³, a = 4x, because (4x)³ = 64x³.

Then fill in the pattern. The sign in the small factor matches the sign in the problem. The middle sign in the big factor is the opposite.

🧠 Let's work it out

Factor x³ − 27.

Write both as cubes: x³ = (x)³ and 27 = 3³.
So a = x and b = 3.
Use a³ − b³ = (a − b)(a² + ab + b²).
Plug in: (x − 3)(x² + 3x + 9).

What do you think comes next?

cube: a number times itself three times: 2³ = 8

Finding zeros by factoring

💡 Factor the polynomial, then set each factor equal to zero.

A zero of f is an x-value that makes f(x) = 0. If a product equals zero, one of its factors must be zero.

Pull out any common factor first, like x. Don't drop it: x = 0 is a zero too.

🧠 Let's work it out

Find the zeros of f(x) = x³ − 16x.

Pull out x: x(x² − 16).
Factor the rest: x(x − 4)(x + 4).
Each factor can be zero, so x = 0, 4, and −4.

What do you think comes next?

zero: an x-value that makes the function equal 0