1Cademy - Multiplying $$(b + 3)(2b^2 - 5b + 8)$$ Using the Vertical Method

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The Vertical Method for Multiplying Binomials

Example

Multiplying $(b + 3)(2b^2 - 5b + 8)$ Using the Vertical Method

Multiply $(b + 3)(2b^2 - 5b + 8)$ using the Vertical Method — the same product as the Distributive Property approach, now organized vertically.

Step 1 — Set up vertically: Place the polynomial with more terms (the trinomial) on top and the one with fewer terms (the binomial) on the bottom, because this produces fewer partial products:

$2b^2 - 5b + 8$ $\times \quad b + 3$

Step 2 — Multiply $(2b^2 - 5b + 8)$ by $3$ : Compute $3 \cdot 2b^2 = 6b^2$ , $3 \cdot (-5b) = -15b$ , and $3 \cdot 8 = 24$ . Write the first partial product:

$6b^2 - 15b + 24$

Step 3 — Multiply $(2b^2 - 5b + 8)$ by $b$ : Compute $b \cdot 2b^2 = 2b^3$ , $b \cdot (-5b) = -5b^2$ , and $b \cdot 8 = 8b$ . Write the second partial product, aligning like terms beneath the first:

$2b^3 - 5b^2 + 8b$

Step 4 — Add the partial products: Combine like terms column by column. The $b^3$ column: $2b^3$ . The $b^2$ column: $-5b^2 + 6b^2 = b^2$ . The $b$ column: $8b - 15b = -7b$ . The constant column: $24$ :

$2b^3 + b^2 - 7b + 24$

The result is $2b^3 + b^2 - 7b + 24$ , which matches the Distributive Property approach. When using the Vertical Method with polynomials of different sizes, placing the polynomial with fewer terms on the bottom produces fewer partial products, making the computation easier to organize.

Updated 2026-04-29

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