Multiplying Polynomials Calculator
Use this multiplying polynomials calculator to expand two polynomial expressions, multiply every term by every term, combine like terms, and write the final answer in standard form. Enter two polynomials such as \(2x^2+3x-4\) and \(x^2-5x+6\), then see the product with step-by-step work.
Polynomial multiplication is based on the distributive property. Every term in the first polynomial must multiply every term in the second polynomial. The calculator below handles monomials, binomials, trinomials, and larger polynomial expressions using the same rule students use by hand.
Multiply two polynomials
Type each polynomial using powers like x^2, x^3, and regular signs. Examples: 3x^2+2x-5, x-4, 2x^3-x+7. Use one variable only.
What does multiplying polynomials mean?
Multiplying polynomials means finding the product of two polynomial expressions. A polynomial is a sum of terms where each term has a coefficient multiplied by a variable raised to a nonnegative whole-number power. Examples include \(3x+2\), \(x^2-5x+6\), \(4x^3+x-9\), and \(2x^4-7x^2+1\). When two polynomials are multiplied, every term in the first polynomial must be multiplied by every term in the second polynomial. After all products are written, like terms are combined to create the final simplified polynomial.
The rule is simple, but the work can become long when the polynomials have many terms. For example, multiplying a binomial by a trinomial creates \(2\times3=6\) separate term products before combining like terms. Multiplying two trinomials creates \(3\times3=9\) separate term products. Multiplying two four-term polynomials creates \(4\times4=16\) products. This is why a step-by-step polynomial multiplication calculator can be helpful: it keeps the distribution organized and reduces sign errors.
The final answer is usually written in standard form. Standard form means the terms are arranged from the highest power of the variable to the lowest power. For example, \(5x^4-3x^3+9x^2-2x+7\) is in standard form because the powers decrease from \(4\) to \(0\). Standard form makes the answer easier to read, compare, graph, factor, and use in later algebra steps.
Main polynomial multiplication rule
\[ \left(\sum_{i=0}^{m} a_i x^i\right)\left(\sum_{j=0}^{n} b_j x^j\right) = \sum_{i=0}^{m}\sum_{j=0}^{n} a_i b_j x^{i+j} \]
This formula means that each term \(a_i x^i\) from the first polynomial multiplies each term \(b_j x^j\) from the second polynomial. The coefficients multiply, and the exponents add.
How to use the multiplying polynomials calculator
Enter the first polynomial in the first input box and the second polynomial in the second input box. Use the caret symbol for powers, such as \(x^2\), \(x^3\), or \(x^4\). You can type positive and negative terms, decimals, constants, and missing powers. For example, \(2x^3-x+8\) is valid even though it does not include an \(x^2\)-term. The calculator reads the missing power as a coefficient of zero.
The variable box lets you choose the variable letter. The default is \(x\), which works for most algebra problems. If your problem uses \(t\), \(y\), or \(m\), you can change the variable letter. Use one variable only because this calculator is designed for single-variable polynomial multiplication. For multivariable expressions such as \(xy+2x\), use a symbolic algebra system or expand manually with extra care.
After you press the calculate button, the calculator parses both polynomials, multiplies each term pair, combines like terms, and displays the final product in standard form. It also shows the term-by-term multiplication summary so students can see how the answer was created. This is important because the final answer alone does not teach the process. The steps help you understand the distributive property and verify the result.
Step-by-step process
- Write both polynomials clearly and arrange each one in descending powers if possible.
- Multiply the first term of the first polynomial by every term in the second polynomial.
- Repeat the process for each remaining term in the first polynomial.
- Use the exponent rule \(x^a\cdot x^b=x^{a+b}\) when multiplying variable powers.
- Multiply the numerical coefficients and keep the correct sign.
- Group like terms that have the same variable power.
- Add or subtract the coefficients of like terms.
- Write the final polynomial in standard form from highest degree to lowest degree.
Formula for multiplying polynomial terms
The smallest building block of polynomial multiplication is multiplying one term by another term. A term such as \(3x^2\) has a coefficient \(3\) and a variable power \(x^2\). A term such as \(-5x^4\) has a coefficient \(-5\) and a variable power \(x^4\). To multiply these terms, multiply the coefficients and add the exponents. This gives \((3x^2)(-5x^4)=-15x^6\).
The reason exponents add is that powers represent repeated multiplication. For example, \(x^2\cdot x^4=(x\cdot x)(x\cdot x\cdot x\cdot x)=x^6\). The coefficient rule and exponent rule work together. Coefficients multiply normally, while exponents add only when the same base is being multiplied.
Term multiplication rule
\[ (ax^m)(bx^n)=abx^{m+n} \]
For example:
\[ (4x^3)(-2x^5)=-8x^8 \]
This term rule is the core of the entire calculator. Whether you multiply a monomial by a binomial, a binomial by a trinomial, or two large polynomials, the same term multiplication rule is repeated until every possible term pair has been multiplied.
Distributive property and polynomial multiplication
The distributive property says that multiplication distributes over addition. In its simplest form, \(a(b+c)=ab+ac\). Polynomial multiplication is just the distributive property applied many times. If you multiply \((x+2)(x^2+3x+4)\), the term \(x\) must multiply every term in \(x^2+3x+4\), and the term \(2\) must also multiply every term in \(x^2+3x+4\).
This gives \(x(x^2+3x+4)+2(x^2+3x+4)\). Then distribute again: \(x^3+3x^2+4x+2x^2+6x+8\). Finally, combine like terms to get \(x^3+5x^2+10x+8\). The process may look long, but every step follows the same rule.
The calculator uses this exact idea. It does not guess the answer. It breaks each polynomial into terms, multiplies each pair of terms, adds coefficients for matching exponents, and then writes the result. That is the same structure students should use on paper.
Example of distribution
\[ (x+2)(x^2+3x+4) = x(x^2+3x+4)+2(x^2+3x+4) \] \[ =x^3+3x^2+4x+2x^2+6x+8 \] \[ =x^3+5x^2+10x+8 \]
Worked examples
Example 1: Multiply a binomial by a trinomial
Multiply \((x+3)(x^2-2x+4)\).
\[ (x+3)(x^2-2x+4) = x(x^2-2x+4)+3(x^2-2x+4) \] \[ =x^3-2x^2+4x+3x^2-6x+12 \] \[ =x^3+x^2-2x+12 \]
The final answer is \(x^3+x^2-2x+12\). The like terms \(-2x^2\) and \(3x^2\) combine to \(x^2\), while \(4x\) and \(-6x\) combine to \(-2x\).
Example 2: Multiply a quadratic by a linear polynomial
Multiply \((3x^2-2x+1)(x-5)\).
\[ (3x^2-2x+1)(x-5) \] \[ =3x^2(x-5)-2x(x-5)+1(x-5) \] \[ =3x^3-15x^2-2x^2+10x+x-5 \] \[ =3x^3-17x^2+11x-5 \]
The answer is \(3x^3-17x^2+11x-5\). Notice that \(-2x\cdot(-5)=10x\), so careful sign handling is essential.
Example 3: Multiply two trinomials
Multiply \((x^2+4x+1)(x^2-x+7)\).
\[ x^2(x^2-x+7)+4x(x^2-x+7)+1(x^2-x+7) \] \[ =x^4-x^3+7x^2+4x^3-4x^2+28x+x^2-x+7 \] \[ =x^4+3x^3+4x^2+27x+7 \]
The final product is \(x^4+3x^3+4x^2+27x+7\). This example shows why combining like terms is necessary after multiplication. Several \(x^2\)-terms appear, and they must be combined into one \(x^2\)-term.
Example 4: Multiply a cubic expression by a quadratic expression
Multiply \((2x^3-x+8)(x^2+3)\).
\[ (2x^3-x+8)(x^2+3) \] \[ =2x^3(x^2+3)-x(x^2+3)+8(x^2+3) \] \[ =2x^5+6x^3-x^3-3x+8x^2+24 \] \[ =2x^5+5x^3+8x^2-3x+24 \]
The answer is \(2x^5+5x^3+8x^2-3x+24\). There is no \(x^4\)-term, so the standard form skips from \(x^5\) to \(x^3\). Missing powers are allowed in polynomials.
Multiplying monomials, binomials, and trinomials
A monomial has one term, a binomial has two terms, and a trinomial has three terms. Polynomial multiplication works the same way for all of them. Multiplying a monomial by a polynomial is usually the simplest case because one term distributes across the entire polynomial. Multiplying a binomial by a binomial creates four products, which is often taught as the FOIL method. Multiplying a trinomial by a trinomial creates nine products, so organization becomes more important.
For a monomial product such as \(3x^2(2x^3-5x+7)\), distribute \(3x^2\) to every term. This gives \(6x^5-15x^3+21x^2\). For a binomial product such as \((x+4)(x-9)\), multiply each term in the first binomial by each term in the second binomial. This gives \(x^2-9x+4x-36=x^2-5x-36\).
For trinomials, a table or grid method can help. Place one polynomial across the top and the other down the side. Fill each cell with a term product, then combine like terms. This is the same as distribution, but it gives a visual structure that reduces missed terms.
| Type of multiplication | Number of products before combining | Example | Method |
|---|---|---|---|
| Monomial × binomial | \(1\times2=2\) | \(3x(x+5)\) | Distribute the monomial |
| Binomial × binomial | \(2\times2=4\) | \((x+2)(x+7)\) | FOIL or distribution |
| Binomial × trinomial | \(2\times3=6\) | \((x+1)(x^2+3x+4)\) | Distribute each binomial term |
| Trinomial × trinomial | \(3\times3=9\) | \((x^2+x+1)(x^2-x+2)\) | Grid method or repeated distribution |
Degree of the product
The degree of a polynomial is the highest exponent of the variable with a nonzero coefficient. For example, \(4x^5-2x^3+x-1\) has degree \(5\). When two nonzero polynomials are multiplied, the degree of the product is usually the sum of the degrees of the factors. If one polynomial has degree \(m\) and the other has degree \(n\), the product has degree \(m+n\).
For example, a quadratic polynomial has degree \(2\), and a cubic polynomial has degree \(3\). Their product has degree \(5\). This happens because the leading term of the first polynomial multiplies the leading term of the second polynomial. If the leading terms are \(ax^m\) and \(bx^n\), their product is \(abx^{m+n}\). Since \(a\) and \(b\) are nonzero, the leading product is also nonzero.
Degree rule
\[ \deg(PQ)=\deg(P)+\deg(Q) \]
For nonzero single-variable polynomials \(P(x)\) and \(Q(x)\), the degree of the product is the sum of the degrees.
Special products in polynomial multiplication
Some polynomial multiplication patterns appear often enough that they are worth memorizing. These patterns are not separate from distribution; they are shortcuts that come from repeated distribution. The square of a binomial is one example. The formula \((a+b)^2=a^2+2ab+b^2\) appears in factoring, completing the square, geometry, and quadratic equations.
The square of a difference is similar: \((a-b)^2=a^2-2ab+b^2\). The last term is positive because a negative times a negative is positive. The middle term is negative because both middle products are negative. A common mistake is writing \((a-b)^2=a^2-b^2\), but that is incorrect. The middle term cannot be ignored.
The difference of squares is another important pattern: \((a+b)(a-b)=a^2-b^2\). In this case, the middle terms cancel. For example, \((x^2+5)(x^2-5)=x^4-25\). This pattern appears frequently when factoring and simplifying rational expressions.
| Special product | Formula | Example | Expanded form |
|---|---|---|---|
| Square of a sum | \((a+b)^2=a^2+2ab+b^2\) | \((x+6)^2\) | \(x^2+12x+36\) |
| Square of a difference | \((a-b)^2=a^2-2ab+b^2\) | \((x-4)^2\) | \(x^2-8x+16\) |
| Difference of squares | \((a+b)(a-b)=a^2-b^2\) | \((x^2+3)(x^2-3)\) | \(x^4-9\) |
Common mistakes when multiplying polynomials
The first common mistake is missing terms during distribution. If a polynomial has three terms and the other polynomial has two terms, there should be six products before like terms are combined. If you only have four products, something was skipped. Counting the expected number of products is a quick way to check your work.
The second common mistake is adding exponents incorrectly. When multiplying powers with the same base, add the exponents. For example, \(x^2\cdot x^5=x^7\), not \(x^{10}\). Exponents multiply only in a power-to-a-power situation, such as \((x^2)^5=x^{10}\). Polynomial multiplication usually uses the product rule for exponents, so the exponents add.
The third common mistake is losing negative signs. If a term is negative, that sign must travel with the coefficient. For example, \((-2x)(-5x^2)=10x^3\), while \((-2x)(5x^2)=-10x^3\). A single sign error can change several terms in the final polynomial.
The fourth common mistake is combining terms that are not like terms. Terms are like terms only if they have the exact same variable power. For example, \(3x^2\) and \(-5x^2\) are like terms, but \(3x^2\) and \(-5x\) are not. You can combine coefficients only when the variable part matches exactly.
The fifth common mistake is leaving the answer unsimplified. After distribution, an expression such as \(x^3+2x^2+5x^2+10x\) should become \(x^3+7x^2+10x\). The calculator shows the product in standard form so the final answer is simplified.
Practice problems
Try these problems after using the calculator. Expand each product and write the answer in standard form.
- \((x+2)(x^2+3x+4)\)
- \((2x-1)(x^2+5x-6)\)
- \((x^2+2x+3)(x^2-x+1)\)
- \((3x^2-4x+2)(2x+5)\)
- \((x^3-2x+7)(x^2+1)\)
- \((2x^2+3x-1)(x^2-4x+8)\)
Answers: \(1)\ x^3+5x^2+10x+8\). \(2)\ 2x^3+9x^2-17x+6\). \(3)\ x^4+x^3+2x^2-x+3\). \(4)\ 6x^3+7x^2-16x+10\). \(5)\ x^5-x^3+7x^2-2x+7\). \(6)\ 2x^4-5x^3+3x^2+28x-8\).
Multiplying polynomials FAQs
What is the rule for multiplying polynomials?
Multiply every term in the first polynomial by every term in the second polynomial, then combine like terms. The term rule is \((ax^m)(bx^n)=abx^{m+n}\).
How do you multiply a polynomial by a monomial?
Distribute the monomial to every term in the polynomial. For example, \(3x(2x^2-5x+4)=6x^3-15x^2+12x\).
How do you multiply two trinomials?
Multiply each of the three terms in the first trinomial by each of the three terms in the second trinomial. This creates nine products before combining like terms.
Do exponents add or multiply when multiplying polynomials?
Exponents add when multiplying powers with the same base. For example, \(x^2\cdot x^3=x^5\). Exponents multiply only when raising a power to another power, such as \((x^2)^3=x^6\).
What is standard form for a polynomial product?
Standard form lists terms in descending powers of the variable, such as \(4x^5-2x^3+x-9\). Missing powers are allowed when their coefficients are zero.
Can this calculator multiply binomials too?
Yes. A binomial is a polynomial with two terms, so this calculator can multiply binomials, trinomials, monomials, and larger single-variable polynomials.
Related Num8ers resources
Polynomial multiplication connects to binomial multiplication, binomial coefficients, polynomial expressions, factoring, and polynomial equations. Use these related resources to support the page after publishing.