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You might know the classic FOIL shortcut for multiplying two binomials, but FOIL only gives you 4 multiplications. What happens when one of your expressions has three terms?
A binomial is a polynomial with two terms, and a trinomial is a polynomial with three terms. To multiply them, every single term in the binomial must multiply every term in the trinomial, producing exactly 2×3=6 terms before combining.
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To guarantee all 6 partial products are accounted for, mathematicians use systematic organization methods such as the horizontal distributive method, a 2×3 area model (grid/box method), or vertical alignment.
The Distributive Method
We use the Extended Distributive Property: every term of the trinomial must be multiplied by every term of the binomial, resulting in 3×2=6 partial product terms prior to combining like terms.
Let's multiply (x+4)(x2+3x−5). We rewrite the problem as x(x2+3x−5)+4(x2+3x−5).
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When multiplying individual terms, use the Product Rule of Exponents: (a⋅xm)⋅(b⋅xn)=(a⋅b)xm+n. You multiply the numerical coefficients and add the exponents of like bases.
When multiplying terms, use the Product Rule of exponents: multiply numerical coefficients and add exponents, (a⋅xm)⋅(b⋅xn)=(a⋅b)xm+n.
Multiplying the first group gives x3+3x2−5x, and multiplying the second group gives 4x2+12x−20.
Now combine like terms—terms that share the exact same variable raised to the same power. Adding the squared terms (3x2+4x2=7x2) and linear terms (−5x+12x=7x) gives our simplified answer: x3+7x2+7x−20.
What happens when higher coefficients and multiple minus signs are involved?