Monomial Times Binomial

Multiply: $3x(2x + 5)$

Distribute the monomial to each term: $3x \times 2x + 3x \times 5$ = Apply distributive property

Multiply the first product: $3x \times 2x = 6x^2$ = $6x^2$

Multiply the second product: $3x \times 5 = 15x$ = $15x$

Combine the terms: $6x^2 + 15x$ = Final answer

Answer: $3x(2x + 5) = 6x^2 + 15x$

FOIL Method with Binomials

Multiply: $(x + 3)(x + 4)$

Apply FOIL - First terms: $x \times x = x^2$ = $x^2$

Outer terms: $x \times 4 = 4x$ = $4x$

Inner terms: $3 \times x = 3x$ = $3x$

Last terms: $3 \times 4 = 12$ = $12$

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

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

Binomials with Subtraction

Multiply: $(2x - 3)(x + 5)$

First terms: $2x \times x = 2x^2$ = $2x^2$

Outer terms: $2x \times 5 = 10x$ = $10x$

Inner terms: $-3 \times x = -3x$ = $-3x$

Last terms: $-3 \times 5 = -15$ = $-15$

Combine like terms: $2x^2 + 10x - 3x - 15$ = $2x^2 + 7x - 15$

Answer: $(2x - 3)(x + 5) = 2x^2 + 7x - 15$

Squaring a Binomial

Expand: $(x + 4)^2$

Rewrite as multiplication: $(x + 4)^2 = (x + 4)(x + 4)$ = Two identical binomials

First terms: $x \times x = x^2$ = $x^2$

Outer + Inner terms: $x \times 4 + 4 \times x = 4x + 4x = 8x$ = $8x$

Last terms: $4 \times 4 = 16$ = $16$

Combine all terms: $x^2 + 8x + 16$ = Perfect square trinomial

Answer: $(x + 4)^2 = x^2 + 8x + 16$

Binomial Times Trinomial

Multiply: $(x + 2)(x^2 - 3x + 1)$

Distribute x to each term: $x(x^2 - 3x + 1) = x^3 - 3x^2 + x$ = $x^3 - 3x^2 + x$

Distribute 2 to each term: $2(x^2 - 3x + 1) = 2x^2 - 6x + 2$ = $2x^2 - 6x + 2$

Combine both results: $x^3 - 3x^2 + x + 2x^2 - 6x + 2$ = All terms written

Group like terms: $x^3 + (-3x^2 + 2x^2) + (x - 6x) + 2$ = Organize by degree

Simplify: $x^3 - x^2 - 5x + 2$ = Final answer

Answer: $(x + 2)(x^2 - 3x + 1) = x^3 - x^2 - 5x + 2$

Mistake: Only multiplying the first and last terms: $(x + 3)(x + 4) = x^2 + 12$

Why: Every term in the first polynomial must multiply every term in the second polynomial.

Correct: Use FOIL: $x^2 + 4x + 3x + 12 = x^2 + 7x + 12$

Mistake: Forgetting the middle term when squaring: $(x + 5)^2 = x^2 + 25$

Why: Squaring a binomial creates three terms, not two. The middle term is $2ab$.

Correct: $(x + 5)^2 = x^2 + 10x + 25$

Mistake: Sign errors with negative terms: $(-3)(5) = 15$

Why: A negative times a positive is negative.

Correct: $(-3)(5) = -15$

Mistake: Adding exponents incorrectly: $x \times x = 2x$

Why: When multiplying variables, we add exponents: $x^1 \times x^1 = x^2$

Correct: $x \times x = x^2$

Area of a Garden

Finding areas with variable dimensions requires polynomial multiplication.

A rectangular garden has length $(x + 5)$ meters and width $(x + 3)$ meters. The area is $(x + 5)(x + 3) = x^2 + 8x + 15$ square meters.

Business Revenue

Revenue calculations often involve multiplying expressions for price and quantity.

If price is $(20 - x)$ dollars and quantity sold is $(100 + 5x)$ items, revenue is $(20 - x)(100 + 5x) = 2000 + 100x - 100x - 5x^2 = 2000 - 5x^2$ dollars.

Q: Does FOIL work for all polynomial multiplication?

A: FOIL only works for multiplying two binomials. For other cases, use the general distributive property.

Q: Why is $(x + 3)^2$ not equal to $x^2 + 9$?

A: Squaring means multiplying by itself: $(x + 3)(x + 3)$. Using FOIL gives $x^2 + 3x + 3x + 9 = x^2 + 6x + 9$.

Q: How do I know when to stop combining terms?

A: Combine terms with the same variable and exponent. Stop when no more like terms remain.