Multiplying Polynomials

Multiplying polynomials builds directly on two skills you already have: the distributive property and combining like terms. Whether you are multiplying a monomial by a polynomial, two binomials, or larger expressions, the underlying strategy is always the same — every term in the first polynomial must be multiplied by every term in the second polynomial, then you combine like terms.

This page covers three approaches: the distributive property (the universal method), the FOIL method (a shortcut for two binomials), and the area model (a visual method that works for any size).

Multiplying a Monomial by a Polynomial

Use the distributive property to multiply the single term by each term inside the polynomial.

Example 1: $3x(4x^2 - 5x + 2)$

Distribute $3x$ to each term:

$3x \cdot 4x^2 = 12x^3$

$3x \cdot (-5x) = -15x^2$

$3x \cdot 2 = 6x$

Combine:

$12x^3 - 15x^2 + 6x$

Answer: $12x^3 - 15x^2 + 6x$

Example 2: $-2x^2(x^3 + 7x - 4)$

$-2x^2 \cdot x^3 = -2x^5$

$-2x^2 \cdot 7x = -14x^3$

$-2x^2 \cdot (-4) = 8x^2$

Answer: $-2x^5 - 14x^3 + 8x^2$

The Distributive Property for Two Polynomials

When multiplying two polynomials, each term in the first polynomial must be multiplied by each term in the second polynomial. Then combine like terms.

Example 3: $(x + 3)(x^2 - 2x + 5)$

Distribute each term of $(x + 3)$ to the entire second polynomial:

Distribute $x$:

$x \cdot x^2 = x^3, \quad x \cdot (-2x) = -2x^2, \quad x \cdot 5 = 5x$

Distribute $3$:

$3 \cdot x^2 = 3x^2, \quad 3 \cdot (-2x) = -6x, \quad 3 \cdot 5 = 15$

Combine all terms:

$x^3 - 2x^2 + 5x + 3x^2 - 6x + 15$

Group and simplify like terms:

$x^3 + (-2x^2 + 3x^2) + (5x - 6x) + 15$

$x^3 + x^2 - x + 15$

Answer: $x^3 + x^2 - x + 15$

The FOIL Method (Two Binomials Only)

FOIL is a shortcut for multiplying two binomials. It stands for First, Outer, Inner, Last — the four products you need:

For $(a + b)(c + d)$:

• First: $a \cdot c$
• Outer: $a \cdot d$
• Inner: $b \cdot c$
• Last: $b \cdot d$

Example 4: $(x + 4)(x + 6)$

• F: $x \cdot x = x^2$
• O: $x \cdot 6 = 6x$
• I: $4 \cdot x = 4x$
• L: $4 \cdot 6 = 24$

Combine:

$x^2 + 6x + 4x + 24 = x^2 + 10x + 24$

Answer: $x^2 + 10x + 24$

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

• F: $3x \cdot 2x = 6x^2$
• O: $3x \cdot 5 = 15x$
• I: $(-2) \cdot 2x = -4x$
• L: $(-2) \cdot 5 = -10$

Combine:

$6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10$

Answer: $6x^2 + 11x - 10$

Example 6: $(5x - 3)(x - 7)$

• F: $5x \cdot x = 5x^2$
• O: $5x \cdot (-7) = -35x$
• I: $(-3) \cdot x = -3x$
• L: $(-3)(-7) = 21$

Combine:

$5x^2 - 35x - 3x + 21 = 5x^2 - 38x + 21$

Answer: $5x^2 - 38x + 21$

The Area Model (Box Method)

The area model works for any polynomial multiplication, not just binomials. Draw a grid with rows for each term of the first polynomial and columns for each term of the second.

Example 7: $(2x + 3)(x + 5)$ Using the Area Model

Collect all four cells: $2x^2 + 10x + 3x + 15$

Combine like terms: $2x^2 + 13x + 15$

Answer: $2x^2 + 13x + 15$

Example 8: $(x + 2)(x^2 + 3x - 1)$ Using the Area Model

Collect: $x^3 + 3x^2 + (-x) + 2x^2 + 6x + (-2)$

Combine like terms: $x^3 + (3x^2 + 2x^2) + (-x + 6x) - 2$

$x^3 + 5x^2 + 5x - 2$

Answer: $x^3 + 5x^2 + 5x - 2$

The area model is especially helpful because it organizes all the partial products visually, making it harder to miss a term.

Multiplying Larger Polynomials

For polynomials with three or more terms on each side, use the distributive property systematically or the area model. FOIL does not apply (it only works for binomial times binomial).

Example 9: $(x^2 + 2x - 1)(x^2 - 3x + 4)$

Distribute each term of the first polynomial across the second:

Distribute $x^2$:

$x^2 \cdot x^2 = x^4, \quad x^2 \cdot (-3x) = -3x^3, \quad x^2 \cdot 4 = 4x^2$

Distribute $2x$:

$2x \cdot x^2 = 2x^3, \quad 2x \cdot (-3x) = -6x^2, \quad 2x \cdot 4 = 8x$

Distribute $-1$:

$(-1) \cdot x^2 = -x^2, \quad (-1) \cdot (-3x) = 3x, \quad (-1) \cdot 4 = -4$

Combine all nine terms:

$x^4 - 3x^3 + 4x^2 + 2x^3 - 6x^2 + 8x - x^2 + 3x - 4$

Group like terms:

$x^4 + (-3x^3 + 2x^3) + (4x^2 - 6x^2 - x^2) + (8x + 3x) - 4$

$x^4 - x^3 - 3x^2 + 11x - 4$

Answer: $x^4 - x^3 - 3x^2 + 11x - 4$

Checking Your Work

A quick way to check polynomial multiplication is to substitute a simple value (like $x = 1$ or $x = 2$) into both the original factors and the product. They should give the same number.

Check Example 4: $(x + 4)(x + 6) = x^2 + 10x + 24$

Let $x = 1$: Left side: $(1 + 4)(1 + 6) = 5 \times 7 = 35$. Right side: $1 + 10 + 24 = 35$. Correct.

This is not a formal proof, but it catches most arithmetic errors.

Real-World Application: Carpentry — Area of a Deck with Border

A carpenter is building a rectangular deck that measures $(x + 6)$ feet by $(x + 4)$ feet, where $x$ represents a variable measurement that depends on the customer’s yard. The total area of the deck is:

$A = (x + 6)(x + 4)$

Using FOIL:

• F: $x \cdot x = x^2$
• O: $x \cdot 4 = 4x$
• I: $6 \cdot x = 6x$
• L: $6 \cdot 4 = 24$

$A = x^2 + 4x + 6x + 24 = x^2 + 10x + 24$

If the base measurement is $x = 10$ feet:

$A = 100 + 100 + 24 = 224 \text{ square feet}$

The carpenter can also verify: $(10 + 6)(10 + 4) = 16 \times 14 = 224$ square feet. The polynomial formula lets the carpenter compute area for any value of $x$ without starting over.

Real-World Application: HVAC — Ductwork Cross-Section

An HVAC technician is designing a rectangular duct with width $(2x + 1)$ inches and height $(x + 3)$ inches. The cross-sectional area determines airflow capacity:

$A = (2x + 1)(x + 3)$

Using FOIL:

• F: $2x \cdot x = 2x^2$
• O: $2x \cdot 3 = 6x$
• I: $1 \cdot x = x$
• L: $1 \cdot 3 = 3$

$A = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3$

For a duct where $x = 5$ inches: $A = 2(25) + 7(5) + 3 = 50 + 35 + 3 = 88$ square inches.

Answer: The cross-sectional area is 88 square inches, which the technician uses to verify the duct meets the required CFM (cubic feet per minute) specifications.

Common Mistakes to Avoid

1. Missing terms in the distribution. Every term in the first polynomial must multiply every term in the second. For a binomial times a trinomial, you should have $2 \times 3 = 6$ partial products before combining.
2. Using FOIL for non-binomials. FOIL only works for two binomials. For anything larger, use the distributive property or area model.
3. Forgetting to combine like terms. After multiplying, you often have several terms with the same degree. Always collect and simplify them.
4. Sign errors with negative terms. Pay close attention to signs: $(-3) \cdot (-7) = +21$ and $(-3) \cdot 7 = -21$. Write out each partial product carefully.
5. Exponent errors. Remember that $x \cdot x^2 = x^3$ (add exponents), not $x^2$.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Multiplying polynomials means every term in the first polynomial multiplies every term in the second
• The distributive property is the universal method — it works for polynomials of any size
• FOIL (First, Outer, Inner, Last) is a shortcut that works only for two binomials
• The area model (box method) organizes partial products visually and works for any size
• After multiplying, always combine like terms and write the answer in standard form
• Check your work by substituting a simple value like $x = 1$ into both the factors and the result

Return to Algebra for more topics in this section.