What Does Multiplying Polynomials Mean?
Multiplying polynomials is the operation of finding the product of two or more polynomials by multiplying each term of one by each term of the other and adding the results. A polynomial is an expression built from variables and coefficients using addition, subtraction, and whole-number powers, such as $2x^2 + 3x - 1$.
The single governing rule is the distributive property: $a(b + c) = ab + ac$, applied repeatedly until every term has met every other term.
This is narrower than the general multiplication of algebraic expressions, which also covers expressions with fractional or negative powers. Polynomials keep whole-number exponents, so the term-by-term product always lands as another polynomial. The related expression, term, factor and coefficient vocabulary is worth having ready.
How Do You Multiply Polynomials?
Two exponent-and-coefficient rules carry every method:
Coefficients multiply: the number parts multiply straight across, so $3 \times 5 = 15$.
Like variables add exponents: $x^2 \times x^3 = x^{5}$, following the exponent rules.
So $3x^2 \times 5x^3 = 15x^{5}$: multiply 3 and 5, add the powers 2 and 3.
Is FOIL different from the distributive property? No. FOIL is just the distributive property with a memory order for the special case of two binomials; it does not extend to longer polynomials, which is a common point of confusion. The distributive idea works for any two polynomials.
Methods for Multiplying Polynomials
Four named methods appear in textbooks, and all four are the distributive property wearing different layouts. Pick by the size of the polynomials, not by preference.
Distributive method: the general rule. Multiply each term of one polynomial by every term of the other, then combine like terms. Works for any two polynomials of any length.
FOIL (First, Outer, Inner, Last): an ordering trick for two binomials only, such as $(x + 4)(x + 5)$. It names the four products so none is dropped; it does not extend past binomials.
Box (grid) method: draw a grid with one polynomial's terms down the side and the other's across the top, fill each cell with a product, then add the cells. Best when either polynomial has three or more terms.
Vertical (column) method: stack the polynomials like a long-multiplication sum, multiply row by row, and add aligned like terms in columns. Handy for large products where keeping columns straight prevents slips.
Each method is worked in full in the examples below.
Examples of Multiplying Polynomials
The set builds from a monomial product up to a trinomial-by-binomial. Each step sits on its own line.
Example 1
Multiply the monomials $4x^3 \times 2x^2$.
Multiply coefficients: $4 \times 2 = 8$.
Add exponents of $x$: $x^3 \times x^2 = x^{5}$.
Final answer: $8x^{5}$.
Example 2
Multiply $(x + 4)(x + 5)$.
The tempting shortcut is to multiply the first terms and the last terms only, writing $x^2 + 20$.
Check it against the area model: a rectangle of sides $x+4$ and $x+5$ has four regions, not two, so two pieces are missing.
The dropped pieces are the outer and inner products $5x$ and $4x$.
Use the full distributive product instead.
$(x + 4)(x + 5) = x \cdot x + x \cdot 5 + 4 \cdot x + 4 \cdot 5$
$= x^2 + 5x + 4x + 20$
$= x^2 + 9x + 20$
Final answer: $x^2 + 9x + 20$. The middle term $9x$ is exactly what the shortcut skipped.
Example 3
Multiply $(2x + 3)(4x + 5)$ using FOIL.
First: $2x \times 4x = 8x^2$
Outer: $2x \times 5 = 10x$
Inner: $3 \times 4x = 12x$
Last: $3 \times 5 = 15$
Combine like terms: $10x + 12x = 22x$.
Final answer: $8x^2 + 22x + 15$.
Example 4
Multiply $3y(5x + 2z)$, a monomial by a binomial.
Distribute $3y$ across both terms.
$3y \times 5x = 15xy$
$3y \times 2z = 6yz$
Final answer: $15xy + 6yz$.
Example 5
Multiply $(x + 2)(x^2 + 3x + 4)$ using the box method.
Draw a grid with $x, 2$ down the side and $x^2, 3x, 4$ across the top, then fill each cell with a product.
$x \cdot x^2 = x^3$, $\quad x \cdot 3x = 3x^2$, $\quad x \cdot 4 = 4x$
$2 \cdot x^2 = 2x^2$, $\quad 2 \cdot 3x = 6x$, $\quad 2 \cdot 4 = 8$
Add the cells and combine like terms: $3x^2 + 2x^2 = 5x^2$ and $4x + 6x = 10x$.
Final answer: $x^3 + 5x^2 + 10x + 8$.
Example 6
Multiply $(2x - 3)(x^2 - x + 5)$ using the vertical method.
Multiply each term of the second polynomial by $2x$, then by $-3$, lining up like terms.
$2x \cdot (x^2 - x + 5) = 2x^3 - 2x^2 + 10x$
$-3 \cdot (x^2 - x + 5) = -3x^2 + 3x - 15$
Add the two rows column by column.
$2x^3 + (-2x^2 - 3x^2) + (10x + 3x) - 15$
$= 2x^3 - 5x^2 + 13x - 15$
Final answer: $2x^3 - 5x^2 + 13x - 15$.
Properties of Multiplying Polynomials
Because the operation is built from ordinary multiplication and addition, it inherits their structure. These properties are what let you check work and reorder steps safely.
Closure: the product of two polynomials is always another polynomial. Whole-number exponents stay whole-number, so nothing "leaves" the polynomial family.
Commutative: order does not matter, $P \cdot Q = Q \cdot P$. So $(x + 2)(x + 5)$ and $(x + 5)(x + 2)$ expand to the same trinomial.
Associative: grouping does not matter, $(P \cdot Q) \cdot R = P \cdot (Q \cdot R)$. When multiplying three polynomials, multiply any two first.
Distributive over addition: $P \cdot (Q + R) = P \cdot Q + P \cdot R$. This is the rule every method rests on.
Degrees add: the degree of the product equals the sum of the degrees, so a degree-2 times a degree-3 polynomial gives a degree-5 result, a fast sanity check on your final answer.
Why Multiplying Polynomials Is the Engine Behind Formulas
Polynomial multiplication exists to answer a practical question: what happens when two changing quantities combine? Area is length times width, and when both are expressions, their product is a polynomial.
The historical thread: systematic term-by-term multiplication traces to the symbolic algebra of François Viète in the late 1500s, who replaced word-problem prose with letters that could be multiplied like numbers.
Where it shows up: projectile motion, compound interest, and every quadratic and cubic polynomial model are polynomial products written out and simplified.
The destination: expanding products is the forward direction; factoring is the reverse, and mastering the expansion is what makes factoring readable later.
The reach is worth seeing now: the same distributive rule that expands $(x+3)(x+2)$ scales up to multiply matrices and to expand the squaring a trinomial identity $(a+b+c)^2$.
Where Multiplying Polynomials Goes Sideways
Most errors are exponent slips or dropped terms, and both are easy to catch with one habit: count your terms.
Mistake 1: Multiplying exponents instead of adding them
Where it slips in: On the first step of any product with powers.
Don't do this: Write $x^2 \times x^3 = x^{6}$.
The correct way: Exponents of like bases add, so $x^2 \times x^3 = x^{5}$. Multiplication of the powers belongs to $(x^2)^3$, a different operation. The first instinct is to multiply the powers because the bases are being multiplied, and separating those two rules is what stops the error.
Mistake 2: Distributing a square across a sum
Where it slips in: Squaring a binomial like $(x + 4)^2$.
Don't do this: Write $(x + 4)^2 = x^2 + 16$.
The correct way: Rewrite the square as a product and multiply it out: $(x + 4)^2 = (x + 4)(x + 4) = x^2 + 8x + 16$. The habit that fixes this is refusing to square term by term; expand the bracket first.
Mistake 3: Forgetting to combine like terms
Where it slips in: After a correct term-by-term expansion, the answer is left sprawling.
Don't do this: Stop at $x^2 + 5x + 4x + 20$.
The correct way: Collect like terms: $5x + 4x = 9x$, giving $x^2 + 9x + 20$. A tidy answer also makes the next factoring step readable.
Conclusion
Multiplying polynomials means multiplying every term of one by every term of the other, then combining like terms.
Coefficients multiply and like-base exponents add.
FOIL, the box method, and the vertical method are all forms of the distributive property.
The frequent mistakes are multiplying exponents, distributing a square across a sum, and skipping the combine-like-terms step.
Expanding products is the reverse of factoring, which makes it a key setup skill.
To practise these methods with a teacher, explore Bhanzu's algebra tutor, get help with algebra, or browse math classes online.
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