Polynomials and Factoring

What's a polynomial?

A polynomial is a sum of terms of the form $a x^n$, where the exponents are non-negative integers.

$p(x) = 4x^3 - 7x^2 + 2x - 5$

• The degree is the highest exponent (here, 3).
• The leading coefficient is the coefficient of the highest-degree term (here, 4).

Adding, subtracting, multiplying

Add/subtract by combining like terms (same variable, same exponent).

$(3x^2 + 5x - 2) + (x^2 - 4x + 6) = 4x^2 + x + 4$

Multiply with the distributive property — every term in the first polynomial times every term in the second.

$(x + 3)(x - 5) = x^2 - 5x + 3x - 15 = x^2 - 2x - 15$

For larger products you can use a grid; for binomials, FOIL works.

Factoring: always start by pulling out a GCF

The greatest common factor is your first move every time.

$6x^3 + 9x^2 = 3x^2(2x + 3)$

Factoring a quadratic $x^2 + bx + c$

Find two numbers that multiply to $c$ and add to $b$.

Example. $x^2 + 7x + 12$. Two numbers with product $12$ and sum $7$: $3$ and $4$.

$x^2 + 7x + 12 = (x + 3)(x + 4)$

For $ax^2 + bx + c$ with $a \neq 1$, use grouping or trial and error.

Special products to recognize on sight

Pattern	Factored
$a^2 - b^2$	$(a - b)(a + b)$
$a^2 + 2ab + b^2$	$(a + b)^2$
$a^2 - 2ab + b^2$	$(a - b)^2$
$a^3 - b^3$	$(a - b)(a^2 + ab + b^2)$
$a^3 + b^3$	$(a + b)(a^2 - ab + b^2)$

Difference of squares. $x^2 - 25 = (x - 5)(x + 5)$.

Perfect square trinomial. $x^2 + 6x + 9 = (x + 3)^2$.

Factoring by grouping

Useful for four-term polynomials.

$x^3 + 2x^2 + 3x + 6$

Group: $(x^3 + 2x^2) + (3x + 6) = x^2(x + 2) + 3(x + 2)$.

Both groups share $(x + 2)$, so:

$= (x + 2)(x^2 + 3)$

Key takeaways

• Combine like terms when adding; distribute when multiplying.
• Always factor out the GCF first.
• For $x^2 + bx + c$, find two numbers with product $c$ and sum $b$.
• Memorize the special-product patterns — difference of squares is the biggest time-saver.