Multiplying a Polynomial by a Polynomial
Multiplying a Polynomial by a Polynomial
Main Rule
To multiply a polynomial by a polynomial, you need to multiply each term of the first polynomial by each term of the second polynomial, and then add all the resulting products.
General Formula
(a + b)(c + d) = ac + ad + bc + bd
Each term of the first parentheses is multiplied by each term of the second parentheses:
- a × c = ac
- a × d = ad
- b × c = bc
- b × d = bd
Distributive Property
Multiplying polynomials is based on the distributive property:
(a + b)(c + d) =
= a(c + d) + b(c + d) =
= ac + ad + bc + bd
The first polynomial is "distributed" across the second: each of its terms multiplies the entire second polynomial.
Counting the Number of Products
Rule: If the first polynomial has m terms and the second has n terms, then there will be a total of m × n products (before combining like terms).
Binomial × Binomial
2 × 2 = 4
products
Binomial × Trinomial
2 × 3 = 6
products
Trinomial × Trinomial
3 × 3 = 9
products
This helps you check that you found all the products!
Step-by-Step Algorithm
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Step 1Multiply the first term of the first polynomial by all terms of the second polynomial
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Step 2Multiply the second term of the first polynomial by all terms of the second polynomial
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Step 3Continue for all remaining terms of the first polynomial
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Step 4Combine like terms (add terms with the same variables)
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Step 5Write the answer in standard form (in descending order of degrees)
(x + 3)(x + 5)
= x·x + x·5 + 3·x + 3·5
← 4 products
= x² + 5x + 3x + 15
← like terms
= x² + 8x + 15
← answer