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Multiplying a Polynomial by a Monomial
To multiply a polynomial by a monomial, apply the Distributive Property: multiply the monomial factor by every term of the polynomial individually, then simplify each resulting product. This operation connects two earlier skills — the Distributive Property and monomial multiplication.
When the outside factor is a simple constant (such as ), each distributed multiplication involves only arithmetic. When the outside factor is a variable monomial (such as , , or ), each multiplication becomes a monomial-times-monomial product that may require multiplying coefficients and adding exponents via the Product Property for Exponents.
The procedure works the same way regardless of how many terms the polynomial contains — the monomial is distributed to every term, whether the polynomial is a binomial (two terms), a trinomial (three terms), or larger.
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Simplifying Using the Distributive Property
Simplifying Using the Distributive Property
Simplifying Using the Distributive Property
Simplifying Using the Distributive Property
Simplifying Using the Distributive Property
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Simplifying by Distributing and Combining Like Terms
Solving by Distributing and Simplifying
Strategy for Solving Linear Equations
Multiplying a Polynomial by a Monomial
Multiplying Using the Distributive Property
Multiplying Using the Distributive Property
Multiplying Using the Distributive Property
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?
A contractor is calculating the total area of two rooms that have the same width 'w' but different lengths 'l1' and 'l2'. The total area is w(l1 + l2). The contractor calculates the area of each room separately as wl1 + wl2 and then adds them. The mathematical rule that justifies this is the ________ property.
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.
Identifying Budgeting Properties in Procurement
Validating Budgeting Formulas in Logistics
Defining the Distributive Property in Administrative Budgeting
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?
Simplifying Using the Distributive Property
Simplifying Using the Distributive Property
Simplifying 6\left( rac{5}{6}y + rac{1}{2} ight) Using the Distributive Property
Simplifying 12\left( rac{1}{3}n + rac{3}{4} ight) Using the Distributive Property
Simplifying Using the Distributive Property
Simplifying Using the Distributive Property
Simplifying Using the Distributive Property
Simplifying Using the Distributive Property
Strategy for Solving Equations with Fraction or Decimal Coefficients
Learn After
Multiplying a Binomial by a Binomial Using the Distributive Property
In a professional landscaping project, if you multiply the cost per square foot 'c' by the sum of the areas of two lawns '(a1 + a2)', the Distributive Property requires you to multiply 'c' by both 'a1' and 'a2' individually.
When a technician multiplies a monomial by a polynomial, such as in the formula P = 2t(4t^2 + 3), they must simplify the product of the variables. According to the Product Property for Exponents, what is the rule for determining the new exponent of the variable 't'?
A manufacturing engineer is calculating the total surface area of a custom part. The width is represented by the monomial 2x and the length is represented by the polynomial (x^2 + 3x - 4). Arrange the steps in the correct order to multiply the monomial 2x by the polynomial (x^2 + 3x - 4).
Resource Management: Distribution Steps
In technical fields such as construction, logistics, and data analysis, formulas are often used to scale multiple values at once. Match each mathematical term or property with its correct role in the process of multiplying a polynomial by a monomial.
Corporate Billing: Applying Discounts
In an automated billing system, when you apply a single-term service fee (a monomial) to a multi-line itemized invoice (a polynomial), the fundamental rule of multiplication requires that you multiply the service fee by ________ term of the invoice individually before simplifying the final expression.
Standard Operating Procedure: Scaling Algebraic Expressions
In a technical manufacturing guide, a technician is required to scale a material stress formula by multiplying a monomial (such as 5x²) by a polynomial. According to the rules for monomial multiplication, which mathematical operations must be performed on the numerical coefficients and the variable exponents respectively?
In a technical manufacturing guide, a technician is required to scale a material stress formula by multiplying a polynomial (such as 2x² + 5x) by a constant scale factor of 4. According to the rules for multiplying a polynomial by a monomial, what will happen to the exponents of the variables in the resulting formula?