Polynomials are the building blocks of algebra — expressions with terms involving powers of variables. Master the four operations (add, subtract, multiply, factor) and you've unlocked half the SAT Math section.
A polynomial is a sum of terms, each of which is a number times a power of a variable: 3x4+2x2−7x+5. The degree is the highest power: this one is degree 4.
Special names by degree:
Degree 0: constant (5)
Degree 1: linear (2x+1)
Degree 2: quadratic (x2−4)
Degree 3: cubic (x3+x)
Adding and subtracting: combine like terms (terms with the same variable raised to the same power).
(3x2+2x)+(x2−5x)=4x2−3x
When subtracting, distribute the negative to every term: (3x2+2x)−(x2−5x)=3x2+2x−x2+5x=2x2+7x.
Multiplying: distribute every term in the first by every term in the second.
(x+2)(x+3)=x2+3x+2x+6=x2+5x+6
Factoring is multiplying in reverse — turning a polynomial into a product. Three patterns to memorize:
Pattern 1: Difference of squares.a2−b2=(a+b)(a−b).
Example:x2−25=(x+5)(x−5).
Pattern 2: Perfect square trinomials.a2+2ab+b2=(a+b)2 and a2−2ab+b2=(a−b)2.
Example:x2+6x+9=(x+3)2.
Pattern 3: Standard quadratic factoring.x2+bx+c — find two numbers that multiply to c and add to b.
Example:x2+5x+6. Two numbers that multiply to 6, add to 5: 2 and 3. So (x+2)(x+3).
Polynomial division. When dividing p(x) by (x−a), you can either do long division or — better — use the factor theorem: if (x−a) is a factor, then p(a)=0.
Example: Is (x−2) a factor of x3−4x2+x+6? Plug in x=2: 8−16+2+6=0. Yes!
Common SAT setup:"For what values of k does the polynomial have a factor of (x−3)?" → plug 3 in for x, solve for k.
Factor: x2−7x+12.
If (x+4) is a factor of f(x)=x3+2x2+kx−24, what is the value of k?
(3x2+2x)−(x2−5x)=3x2+2x−x2+5x, NOT 3x2+2x−x2−5x. Distribute the minus to every term inside the parentheses. This is the #1 sign error on polynomial subtraction.
(x+3)2=x2+6x+9, NOT x2+9. Always FOIL out perfect squares — don't just square the terms separately. Same trap with (a−b)2=a2−2ab+b2, not a2−b2.
2x2+6x+4: factor out 2 first → 2(x2+3x+2)=2(x+1)(x+2). Trying to factor without pulling out the GCF makes the problem harder than it needs to be.
x4−16 → first factor as difference of squares: (x2−4)(x2+4). But x2−4 is also a difference of squares: (x−2)(x+2). So fully factored: (x−2)(x+2)(x2+4). Always check if any factor can be factored further.
A polynomial is a sum of terms with non-negative integer powers of variables. Degree = highest power.
When subtracting polynomials, distribute the negative to every term inside the parentheses.
Three factoring patterns: difference of squares (a2−b2), perfect square trinomial (a2±2ab+b2), standard quadratic (x2+bx+c).
Factor theorem:(x−a) is a factor of p(x) if and only if p(a)=0.
Always pull out the GCF first, and always check whether a factor can be factored further.