4.1 Polynomial operations & factoring
Key Takeaways
- Combine only like terms; when subtracting, distribute the negative sign to every term inside the parentheses first.
- Add exponents when multiplying like bases (x^a·x^b = x^(a+b)) and multiply them for a power of a power ((x^a)^b = x^(ab)).
- FOIL multiplies two binomials — First, Outer, Inner, Last — then combine the two middle terms.
- Always factor out the GCF first, then check for a difference of squares a²−b² = (a+b)(a−b).
- Factor x²+bx+c by finding two numbers that multiply to c and add to b; use the ac-method for ax²+bx+c.
Polynomials and Their Parts
A polynomial is a sum of terms, where each term is a number (the coefficient) times a variable raised to a whole-number power, such as 3x^2 - 5x + 7. The degree of a polynomial is its largest exponent; here the degree is 2. Terms that share the same variable and the same exponent are like terms, and only like terms may be combined. On the PERT Mathematics subtest you will add, subtract, multiply, and factor these expressions quickly, so the mechanics matter far more than the vocabulary.
Laws of exponents
Multiplying and factoring both depend on the exponent rules, so keep this table handy:
| Rule | Formula | Example |
|---|---|---|
| Product | x^a · x^b = x^(a+b) | x^3 · x^4 = x^7 |
| Quotient | x^a ÷ x^b = x^(a−b) | x^6 ÷ x^2 = x^4 |
| Power of a power | (x^a)^b = x^(ab) | (x^2)^3 = x^6 |
| Power of a product | (xy)^a = x^a·y^a | (2x)^3 = 8x^3 |
| Zero exponent | x^0 = 1 | 7^0 = 1 |
A frequent trap is adding exponents when you should multiply them, or the reverse. Remember: you add exponents when like bases are multiplied, and you multiply exponents when a power is raised to another power.
Adding and subtracting polynomials
To add or subtract polynomials, combine like terms. When subtracting, first distribute the negative sign to every term inside the parentheses.
Example 1. Simplify (4x^2 - 3x + 5) - (x^2 + 6x - 2). Distribute the minus sign across the second group: 4x^2 - 3x + 5 - x^2 - 6x + 2. Now group the like terms: (4x^2 - x^2) + (-3x - 6x) + (5 + 2) = 3x^2 - 9x + 7. The sign slip of leaving +6x instead of -6x is the single most common mistake, so write out the full distribution before you combine anything.
Multiplying with distribution and FOIL
To multiply a monomial by a polynomial, distribute the monomial to each term: 2x(3x - 4) = 6x^2 - 8x. To multiply two binomials, use FOIL — First, Outer, Inner, Last.
Example 2. Multiply (x + 5)(x - 3).
- First:
x · x = x^2 - Outer:
x · (-3) = -3x - Inner:
5 · x = 5x - Last:
5 · (-3) = -15
Combine the two middle terms: x^2 - 3x + 5x - 15 = x^2 + 2x - 15. For a larger product, distribute every term of the first polynomial across the second: (x + 2)(x^2 - 3x + 1) = x^3 - 3x^2 + x + 2x^2 - 6x + 2 = x^3 - x^2 - 5x + 2.
Factoring
Factoring reverses multiplication: you rewrite a polynomial as a product of simpler factors. Always work through the same three steps in order.
- Pull out the GCF (greatest common factor) first.
- Check for a difference of squares.
- Factor the trinomial that remains.
Greatest common factor
Example 3. Factor 6x^3 + 9x^2. The GCF of the coefficients 6 and 9 is 3, and both terms share x^2, so the overall GCF is 3x^2. Pulling it out gives 3x^2(2x + 3). Check by distributing: 3x^2 · 2x = 6x^3 and 3x^2 · 3 = 9x^2. Correct.
Difference of squares
The pattern a^2 - b^2 = (a + b)(a - b) applies whenever two perfect squares are subtracted.
Example 4. Factor x^2 - 49. Here a = x and b = 7, so x^2 - 49 = (x + 7)(x - 7). Note that a sum of squares such as x^2 + 49 does not factor over the real numbers, so leave it alone.
Trinomials of the form x² + bx + c
Find two numbers that multiply to c and add to b.
Example 5. Factor x^2 + 7x + 12. You need two numbers that multiply to 12 and add to 7 — namely 3 and 4 — so x^2 + 7x + 12 = (x + 3)(x + 4). When c is negative, the two numbers have opposite signs; for x^2 - 2x - 15, use -5 and 3 to get (x - 5)(x + 3).
Trinomials of the form ax² + bx + c
When the leading coefficient is not 1, use the ac-method: multiply a · c, find two numbers with that product that also sum to b, split the middle term, and factor by grouping.
Example 6. Factor 2x^2 + 7x + 3. Since a·c = 2·3 = 6, look for two numbers that multiply to 6 and add to 7 — that is 6 and 1. Split the middle term: 2x^2 + 6x + 1x + 3. Group and factor each pair: 2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1). Check with FOIL: 2x^2 + x + 6x + 3 = 2x^2 + 7x + 3. Correct. These factored forms become the solutions to quadratic equations in the next section.
Simplify (3x² + 2x − 1) − (x² − 4x + 6).
Factor completely: 4x² − 25.
Simplify (2x³)².