Polynomials

A1.H.1Add & subtract polynomials01/05

Add and subtract polynomials.

Simplify

(3 x^{2} + 2 x - 1) - (x^{2} - 4 x + 5)

Step01of 05

Adding polynomials = combining like terms. Same variable, same power → can be combined. Different power → can't.

$3 x^{2}$ and $- x^{2}$	→like (both x²)
$2 x$ and $- 4 x$	→like (both x)
$- 1$ and $5$	→like (both constants)
$3 x^{2}$ and $2 x$	→NOT like (different power)

A1.H.1 — Add and subtract polynomials.

Simplify

(3 x^{2} + 2 x - 1) - (x^{2} - 4 x + 5)

Step 01 of 05

Adding polynomials = combining like terms. Same variable, same power → can be combined. Different power → can't.

$3 x^{2}$ and $- x^{2}$	→like (both x²)
$2 x$ and $- 4 x$	→like (both x)
$- 1$ and $5$	→like (both constants)
$3 x^{2}$ and $2 x$	→NOT like (different power)

Step 02 of 05

Step 1 — distribute the minus sign. Subtracting a polynomial means flipping every sign inside the second set of parens. This is where the bulk of errors happen.

(3 x^{2} + 2 x - 1) - (x^{2} - 4 x + 5)

= 3 x^{2} + 2 x - 1 - x^{2} + 4 x - 5

Notice every sign in the second polynomial flipped: $+ x^{2} \to - x^{2}$ , $- 4 x \to + 4 x$ , $+ 5 \to - 5$ .

Step 03 of 05

Step 2 — group like terms together. Slide them next to each other (commutative property earns its keep again).

= (3 x^{2} - x^{2}) + (2 x + 4 x) + (- 1 - 5)

Step 04 of 05

Step 3 — combine each group.

= 2 x^{2} + 6 x - 6

Three terms, each from one group. Standard form lists terms in descending power.

Step 05 of 05

For ADDITION (no minus in front of the second polynomial), skip the sign-flip step.

(3 x^{2} + 2 x - 1) + (x^{2} - 4 x + 5)

= 4 x^{2} - 2 x + 4

Key insight

Subtraction = "add the opposite." Distribute the minus across every term of the second polynomial, THEN combine like terms. Same variable, same power = combinable.

A1.H.2 — Multiply binomials — FOIL.

Multiply

(x + 3) (x + 5)

Step 01 of 05

When you multiply two binomials, every term in the first must multiply every term in the second — that's 4 products total. FOIL is the mnemonic for tracking them in order.

First	→ $x \cdot x$ — first term of each
Outer	→ $x \cdot 5$ — outer terms of the whole expression
Inner	→ $3 \cdot x$ — inner terms (next to each other)
Last	→ $3 \cdot 5$ — last term of each

Step 02 of 05

Apply FOIL to $(x + 3) (x + 5)$ .

F: O: I: L: x \cdot x = x^{2} x \cdot 5 = 5 x 3 \cdot x = 3 x 3 \cdot 5 = 15

Step 03 of 05

Sum the four products and combine like terms. The Outer and Inner are usually like terms.

x^{2} + 5 x + 3 x + 15

= x^{2} + 8 x + 15

Step 04 of 05

With negatives — keep signs attached to their numbers. $(x - 4) (x + 2)$ :

F: O: I: L: x \cdot x = x^{2} x \cdot 2 = 2 x (- 4) \cdot x = - 4 x (- 4) \cdot 2 = - 8

= x^{2} + 2 x - 4 x - 8 = x^{2} - 2 x - 8

Step 05 of 05

FOIL ONLY works for binomial × binomial. For more terms (a trinomial times a binomial, say), use the general distributive rule — every term times every term.

(x + 2) (x^{2} - 3 x + 1)

= x \cdot x^{2} + x (- 3 x) + x (1) + 2 x^{2} + 2 (- 3 x) + 2 (1)

= x^{3} - 3 x^{2} + x + 2 x^{2} - 6 x + 2 = x^{3} - x^{2} - 5 x + 2

Key insight

FOIL is just bookkeeping for the distributive property when both factors have exactly two terms. The underlying rule — "every term times every term" — works for any size factors. FOIL just gives you a memorable order for the binomial case.

A1.H.3 — Special products — patterns worth memorizing.

Step 01 of 05

Three FOIL products show up so often they get their own pattern. Memorize them and you'll skip the FOIL bookkeeping every time you spot one.

Square of a sum	→ $(a + b)^{2} = a^{2} + 2 a b + b^{2}$
Square of a difference	→ $(a - b)^{2} = a^{2} - 2 a b + b^{2}$
Difference of squares	→ $(a + b) (a - b) = a^{2} - b^{2}$

Step 02 of 05

Square of a sum. The middle term comes from the Outer + Inner = $2 a b$ .

(x + 4)^{2} = (x + 4) (x + 4)

= x^{2} + 4 x + 4 x + 16 = x^{2} + 8 x + 16

Pattern check: $a = x, b = 4$ , so the result is $x^{2} + 2 (4) x + 16 = x^{2} + 8 x + 16$ $✓$ .

Step 03 of 05

Square of a difference. Same as above, but the middle term has a minus sign — the last term stays positive (negative times negative).

(x - 5)^{2} = x^{2} - 10 x + 25

Common error: writing $(x - 5)^{2} = x^{2} - 25$ . Squaring does NOT distribute over subtraction. The middle term is real and non-negotiable.

Step 04 of 05

Difference of squares. Outer and Inner cancel — no middle term at all.

(x + 7) (x - 7) = x^{2} + 7 x - 7 x - 49 = x^{2} - 49

Spotting it: same first term, opposite signs on the second term. The cancellation is automatic.

Step 05 of 05

Why memorize? These three patterns appear constantly: factoring quadratics, completing the square, simplifying rational expressions, even later in calculus. Recognizing them on sight saves a FOIL pass every single time.

$(x + 6)^{2}$	→ $x^{2} + 12 x + 36$
$(2 x - 3)^{2}$	→ $4 x^{2} - 12 x + 9$
$(5 x + 4) (5 x - 4)$	→ $25 x^{2} - 16$

Key insight

See

()^{2}

→ write three terms (square, twice the product, square). See

(a + b) (a - b)

→ write two terms (square minus square). Both come straight from FOIL — but spotting them lets you skip the work.

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