Basic Binomial Multiplication

Multiply $(x + 2)(x + 3)$

First: Multiply the first terms: $x \cdot x = x^2$ = $x^2$

Outer: Multiply the outer terms: $x \cdot 3 = 3x$ = $3x$

Inner: Multiply the inner terms: $2 \cdot x = 2x$ = $2x$

Last: Multiply the last terms: $2 \cdot 3 = 6$ = $6$

Combine all terms: $x^2 + 3x + 2x + 6$ = $x^2 + 5x + 6$

Answer: $(x + 2)(x + 3) = x^2 + 5x + 6$

Binomials with Subtraction

Multiply $(x + 5)(x - 2)$

First: Multiply the first terms: $x \cdot x = x^2$ = $x^2$

Outer: Multiply the outer terms: $x \cdot (-2) = -2x$ = $-2x$

Inner: Multiply the inner terms: $5 \cdot x = 5x$ = $5x$

Last: Multiply the last terms: $5 \cdot (-2) = -10$ = $-10$

Combine like terms: $x^2 - 2x + 5x - 10 = x^2 + 3x - 10$ = $x^2 + 3x - 10$

Answer: $(x + 5)(x - 2) = x^2 + 3x - 10$

Binomials with Coefficients

Multiply $(2x + 1)(x + 4)$

First: Multiply the first terms: $2x \cdot x = 2x^2$ = $2x^2$

Outer: Multiply the outer terms: $2x \cdot 4 = 8x$ = $8x$

Inner: Multiply the inner terms: $1 \cdot x = x$ = $x$

Last: Multiply the last terms: $1 \cdot 4 = 4$ = $4$

Combine like terms: $2x^2 + 8x + x + 4 = 2x^2 + 9x + 4$ = $2x^2 + 9x + 4$

Answer: $(2x + 1)(x + 4) = 2x^2 + 9x + 4$

Both Terms Negative

Multiply $(x - 3)(x - 4)$

First: Multiply the first terms: $x \cdot x = x^2$ = $x^2$

Outer: Multiply the outer terms: $x \cdot (-4) = -4x$ = $-4x$

Inner: Multiply the inner terms: $(-3) \cdot x = -3x$ = $-3x$

Last: Multiply the last terms: $(-3) \cdot (-4) = 12$ = $12$

Combine like terms: $x^2 - 4x - 3x + 12 = x^2 - 7x + 12$ = $x^2 - 7x + 12$

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

Mistake: Forgetting to multiply ALL four pairs of terms

Why: Students sometimes only multiply the first and last terms, missing the middle terms entirely.

Correct: Always follow F-O-I-L in order: First, Outer, Inner, Last. You should have 4 terms before combining.

Mistake: Sign errors when multiplying negatives

Why: Negative times negative equals positive, but students often forget this rule.

Correct: In $(x - 3)(x - 4)$, the Last step gives $(-3)(-4) = +12$, not $-12$.

Mistake: Not combining like terms at the end

Why: The Outer and Inner terms often produce like terms that should be combined.

Correct: After FOIL, always check for like terms. For $(x+2)(x+3)$: $3x + 2x = 5x$.

Mistake: Confusing $x \cdot x$ with $x + x$

Why: The First step is multiplication, not addition.

Correct: $x \cdot x = x^2$, not $2x$. Remember: multiplying same variables means adding exponents.

Area of a Garden

A rectangular garden has dimensions that can be expressed as binomials.

If a garden is $(x + 3)$ meters long and $(x + 2)$ meters wide, its area is $(x+3)(x+2) = x^2 + 5x + 6$ square meters.

Projectile Motion

In physics, the path of a thrown object often involves products of binomials.

If height depends on $(t + 2)(t - 5)$ where $t$ is time, expanding gives $t^2 - 3t - 10$ which is easier to analyze.

Q: Does FOIL work for multiplying any polynomials?

A: FOIL only works for multiplying two binomials (expressions with exactly 2 terms each). For other polynomials, use the distributive property to multiply each term in one polynomial by every term in the other.

Q: Why do the Outer and Inner terms usually combine?

A: In $(x + a)(x + b)$, the Outer gives $bx$ and the Inner gives $ax$. Both are $x$ terms (like terms), so they combine to $(a + b)x$.

Q: Is there a pattern for $(x + a)(x + b)$?

A: Yes! $(x + a)(x + b) = x^2 + (a+b)x + ab$. The middle coefficient is the sum of $a$ and $b$, and the constant is their product.