Polynomials: Operations and Factoring
Adding, subtracting, multiplying, and breaking down polynomials
Adding, subtracting, multiplying, and breaking down polynomials
A polynomial is a mathematical expression made up of terms added (or subtracted) together. Each term is a number, a variable, or a number multiplied by a variable raised to a power. The expression 3x2 + 2x - 5 is a polynomial. The individual parts — 3x2, 2x, and -5 — are called terms. A polynomial with one term is a monomial (like 4x3). Two terms makes a binomial (like x + 1). Three terms makes a trinomial (like x2 + 3x + 2).
The degree of a polynomial is the highest power of the variable. In 3x4 + 2x2 - 7, the degree is 4. Identifying the degree is important because it tells you the shape of the graph — a degree-2 polynomial (quadratic) makes a parabola, while a degree-3 polynomial makes an S-shaped curve.
Adding and subtracting polynomials comes down to combining like terms — terms that have exactly the same variable part. In 3x2 + 2x2, both terms have x2, so you add the coefficients: 3 + 2 = 5, giving 5x2. You cannot combine terms with different powers: 3x2 and 2x are not like terms and must stay separate.
Example: (2x2 + 3x + 1) + (x2 - 2x + 4) = 2x2 + x2 + 3x - 2x + 1 + 4 = 3x2 + x + 5. Subtraction works the same way — just distribute the negative sign first: (2x2 + 3x) - (x2 + 1) = 2x2 + 3x - x2 - 1 = x2 + 3x - 1.
When multiplying two binomials, use the FOIL method: First, Outer, Inner, Last. For (x + 3)(x + 5): First: x × x = x2. Outer: x × 5 = 5x. Inner: 3 × x = 3x. Last: 3 × 5 = 15. Then combine: x2 + 5x + 3x + 15 = x2 + 8x + 15.
For multiplying a binomial by a trinomial, use the distributive property (also called expanding). Multiply each term in the first parentheses by each term in the second: (x + 2)(x2 + 3x + 4) = x(x2 + 3x + 4) + 2(x2 + 3x + 4) = x3 + 3x2 + 4x + 2x2 + 6x + 8 = x3 + 5x2 + 10x + 8.
Factoring is the reverse of expanding — instead of distributing, you find what was multiplied together to create the polynomial. The first step is always to factor out the greatest common factor. Find the largest number and variable that divides every term. In 4x3 + 8x2, the GCF is 4x2. Dividing each term: 4x3 ÷ 4x2 = x, and 8x2 ÷ 4x2 = 2. So 4x3 + 8x2 = 4x2(x + 2).
Factoring trinomials of the form x2 + bx + c means finding two numbers that multiply to c and add to b. For x2 + 5x + 6, you need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3. So x2 + 5x + 6 = (x + 2)(x + 3). Always check by expanding: (x + 2)(x + 3) = x2 + 3x + 2x + 6 = x2 + 5x + 6. Correct!
For more challenging trinomials like 2x2 + 7x + 3, try the "AC method": multiply a and c (2 × 3 = 6), find two numbers that multiply to 6 and add to 7 (6 and 1), then rewrite and group: 2x2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Some polynomial patterns appear frequently. The most important is the difference of squares: a2 - b2 = (a + b)(a - b). This works because when you expand (a + b)(a - b), the middle terms cancel: a2 - ab + ab - b2 = a2 - b2. Example: x2 - 9 = x2 - 32 = (x + 3)(x - 3). Always look for this pattern — it makes factoring much faster once you recognize it.
Factoring is essential because it helps you solve equations. If you know that x2 - 9 = 0, and you know x2 - 9 = (x + 3)(x - 3) = 0, then either x + 3 = 0 (so x = -3) or x - 3 = 0 (so x = 3). This ability to break down complex expressions into simpler factors is one of the most useful skills in algebra.