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Multiplying Using the Distributive Property
Multiply , where a negative monomial is distributed across a trinomial.
Step 1 — Distribute: Multiply by each of the three terms in the trinomial:
Step 2 — Simplify: Compute each monomial product by multiplying the coefficients and adding exponents:
- and , giving
- and , giving
- (subtracting a negative produces a positive)
The result is . This example combines several skills: distributing a negative monomial, applying sign rules for multiplication, using the Product Property for Exponents, and recognizing that subtracting a negative becomes addition. The procedure is the same whether the polynomial has two terms or three — every term receives the monomial factor.
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Ch.6 Polynomials - Elementary Algebra @ OpenStax
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A payroll specialist is calculating a standard 5 percent bonus for two different departments. The total bonus can be represented by the expression 0.05(S + M), where S is the total salary of the Sales department and M is the total salary of the Marketing department. Which mathematical property justifies rewriting this expression as 0.05S + 0.05M?
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A payroll administrator is expanding the expression 40(r + 2) to calculate total weekly pay including a 2 dollar hourly raise. Arrange the steps in the correct order to expand this expression using the Distributive Property.
An HR manager is simplifying payroll formulas for different departments. Match each initial cost expression on the left with its equivalent simplified form on the right using the distributive property.
A facilities manager is calculating the total cost of upgrading n standard workstations and 5 conference rooms, where the cost per upgrade is x dollars. The total cost is represented by the expression x(n + 5). True or False: According to the distributive property, this expression is equivalent to xn + 5.
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A retail manager is calculating the total revenue from selling 'k' holiday gift baskets. Each basket contains a bottle of sparkling cider (c) and a box of chocolates (h). The total revenue is represented by the expression k(c + h). According to the distributive property, which of the following is equivalent to this expression?
A project coordinator is calculating the total labor cost for two separate teams using the expression (Hours_Team_A + Hours_Team_B) × Hourly_Rate. To allocate expenses to different departments, the coordinator rewrites this as (Hours_Team_A × Hourly_Rate) + (Hours_Team_B × Hourly_Rate). Which mathematical property justifies this expansion?
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A logistics coordinator is simplifying a cost-distribution formula represented by the expression . According to the distributive property, what is the result of the first step: multiplying by the first term, ?
A production analyst is simplifying a cost-adjustment formula: -2y(4y^2 + 3y - 5). Match each term from the trinomial with its correct product after being multiplied by the monomial -2y.
A business analyst is simplifying a growth projection formula represented by the expression . When applying the distributive property to multiply the term by , the exponent of the resulting product is determined by ____ the exponents of the individual y terms.
An operations analyst is simplifying a cost-adjustment formula represented by the expression . True or False: When applying the distributive property, multiplying the negative monomial by the constant term results in a positive product ().
A market analyst is simplifying a consumer engagement model represented by the expression . Arrange the following terms in the correct order as they would appear in the final simplified polynomial expression, starting from the result of the first distribution step.
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A logistics coordinator is using the expression to adjust shipment volumes. When applying the distributive property to the middle term, , what is the coefficient of the resulting product?
A production supervisor is explaining how to simplify a resource-adjustment formula represented by the expression . According to the distributive property, which statement correctly describes how the monomial must be applied to the trinomial?