Rewrite each of the following binomials as the product of an integer with a different binomial

In this explainer, we will learn how to multiply a binomial by a polynomial of order three and similar algebraic expressions.

Multiplying polynomial expressions is an application of two results: the distributive property of multiplication over addition and subtraction and the power rule for exponents.

We can apply the distributive property of multiplication over addition and subtraction since variables represent numbers, which means they must satisfy any properties of numbers. In particular, we know that for any rational numbers 𝑎,𝑏, and 𝑐, we have 𝑎×(𝑏+𝑐)=(𝑎×𝑏)+(𝑎×𝑐 ).

We can use this to expand the product of any two polynomials. However, it is beyond the scope of this explainer. Instead, we will use this to expand the product of a binomial and a trinomial. For example, let’s expand 𝑥−𝑥+3(𝑥+2). We set 𝑎=𝑥−𝑥+3, 𝑏=𝑥, and 𝑐=2 to get

We can now distribute the monomials over the polynomials. We do this by multiplying each term by the factor 𝑥−𝑥+3. Let’s start with the first term:

Now, we want to apply the product rule for exponents, which tells us that for any positive integers 𝑚 and 𝑛, we have 𝑥×𝑥=𝑥. We rewrite 𝑥 as 𝑥 to get 𝑥×𝑥−(𝑥×𝑥)+(3×𝑥)=𝑥×𝑥−𝑥×𝑥+3𝑥=𝑥−𝑥+3𝑥=𝑥−𝑥+3𝑥.

We can apply this same process to the other term: 𝑥−𝑥+3×2=𝑥×2−(𝑥×2)+(3×2)=2𝑥−2𝑥+6.

The sum of these two expressions gives us the product: 𝑥−𝑥+3(𝑥+2)= 𝑥−𝑥+3𝑥+2𝑥−2𝑥+ 6.

We can now rearrange and combine like terms, which we recall are terms with the same variables raised to the same powers.

Hence, 𝑥−𝑥+3(𝑥+2)=𝑥+𝑥+𝑥+6.

We could also do this expansion using an area model. We construct a rectangle with sides of lengths 𝑥−𝑥+3 and 𝑥+2 and determine the expressions for the area of each section as shown.

We can simplify each expression using the commutativity of multiplication and the product rule for exponents. This gives us the following.

We can now add all of these expressions for the area of the smaller rectangles to expand the product:

Let’s now see an example of applying this process to expand and simplify the product of a binomial and a trinomial.

Example 1: Expanding and Simplifying the Product of a Binomial and Trinomial

Expand and simplify the expression (𝑥+2)𝑥−5 𝑥+3.

Answer

We are asked to expand and simplify the product of a binomial and a trinomial. To do this, we could distribute the multiplication of the factor (𝑥+2) over the addition and subtraction in the trinomial; however, it is easier to distribute the other way around. We can do this since the variable 𝑥 is a rational number, so we know it must also satisfy the distributive property.

Distributing 𝑥 −5𝑥+3 over the binomial gives us

Each term on the right-hand side of this equation is the product of a monomial and a trinomial. We can now simplify each term by distributing the monomial over the trinomial and then by using the product rule for exponents.

In the first term, we have

We can simplify the product of 𝑥  and 𝑥 by noting that 𝑥=𝑥 and that the product rule for exponents tells us that for any positive integers 𝑚 and 𝑛, we have 𝑥×𝑥=𝑥. Thus, 𝑥×𝑥−(𝑥×5𝑥)+ (𝑥×3)=𝑥−5𝑥+3𝑥=𝑥−5𝑥+3𝑥. 

We can follow the same process for the other term: 2𝑥−5𝑥+3=2×𝑥−(2×5𝑥)+(2×3)=2𝑥−10𝑥+6. 

The sum of these two expressions then gives us the expanded product of the binomial and trinomial in the question: (𝑥+2)𝑥−5𝑥+3=𝑥−5𝑥+3𝑥+2𝑥−10𝑥+6.

We can finally simplify by combining like terms: 𝑥−5𝑥+3𝑥+2𝑥−10𝑥+6=𝑥+(−5+2)𝑥+(3−10)𝑥+6=𝑥−3𝑥−7𝑥+6.

Hence, (𝑥+2)𝑥−5𝑥+3=𝑥−3𝑥−7𝑥+6.

We can follow this process to expand the product of any binomial and trinomial; however, we may not be given the trinomial in standard form. We could be given the trinomial in a factored form or as a binomial squared. For example, (𝑥+1)(𝑥+2)(𝑥+3), or (𝑥+1)(𝑥+2).

In both of these cases, the process is the same. We apply the associativity property of multiplication to evaluate the product in any order we want. This allows us to multiply two of the binomial factors to get a trinomial. For example, since (𝑥+2)=(𝑥+2)(𝑥+2) =𝑥+4𝑥+4, we have (𝑥+1)(𝑥+2 )=(𝑥+1)𝑥+4𝑥+ 4.

At this stage, we can follow the process of multiplying a binomial by a trinomial to expand and simplify this expression as demonstrated previously.

In our next example, we will follow this process to simplify the product of three binomials.

Example 2: Expanding and Simplifying the Product of Three Binomials

Expand and simplify (𝑥+2)( 𝑥−7)(𝑥+4).

Answer

We are asked to expand and simplify the product of three binomials. To do this, we first note that multiplication is associative. Thus, we can evaluate the product of these binomials by first multiplying two of the binomials together. Let’s expand and simplify (𝑥+2)( 𝑥−7). We can do this by using the horizontal method:

We can then simplify by combining like terms: 𝑥−7𝑥+2𝑥−14=𝑥 +(−7+2)𝑥−14=𝑥 −5𝑥−14.

Thus, (𝑥+2)(𝑥−7 )(𝑥+4)=𝑥−5𝑥− 14(𝑥+4).

We now distribute the factor of 𝑥+4 over the trinomial. This gives us  𝑥−5𝑥−14(𝑥+4)=𝑥(𝑥+4)−5𝑥(𝑥+4)−14(𝑥+4).

To simplify further, we need to distribute each factor over the parentheses. We can do this by recalling the product rule for exponents: 𝑥×𝑥=𝑥 . We can evaluate this for each term separately. We get 𝑥(𝑥+4)=𝑥+4𝑥,−5𝑥(𝑥+4)=−5𝑥−20𝑥,−14(𝑥+4)=−14𝑥−56.

Substituting these into our expression gives 𝑥(𝑥+4)−5𝑥(𝑥+4)−14(𝑥+4)=𝑥+4𝑥−5𝑥−20𝑥−14𝑥−56.

We can now simplify by combining like terms: 𝑥+4𝑥−5𝑥−20𝑥−14𝑥−56=𝑥+(4−5)𝑥+(−20−14)𝑥−56=𝑥−𝑥−34𝑥−56.

Hence, (𝑥+2)(𝑥−7)(𝑥+4)=𝑥−𝑥−34𝑥−56.

In our next example, we will expand and simplify the product of a binomial expression with the square of a binomial expression.

Example 3: Expanding and Simplifying the Product of a Binomial and the Square of a Binomial

Expand and simplify the algebraic expression (𝑥−2)(𝑥+6 ).

Answer

We are asked to expand and simplify the product of a binomial expression with the square of a binomial expression. To do this, we first note that multiplication is associative. Thus, we can evaluate the product of these binomials by first multiplying two of the binomials together. Let’s start by expanding the square of a binomial. We can do this using the horizontal method: (𝑥+6)=(𝑥 +6)(𝑥+6)=𝑥(𝑥+6)+6(𝑥+6)=𝑥+6𝑥+6𝑥+36=𝑥+12𝑥+36.

Thus, (𝑥−2)(𝑥+6)=(𝑥−2)𝑥+12𝑥+36.

We can now distribute the factor of 𝑥−2 over the trinomial. We get

In each term, we have the product of a monomial and a binomial. We can expand each term by distributing the monomial factor over the binomial to get

Now, we simplify each product: 𝑥×𝑥−𝑥× 2+(12𝑥×𝑥)−(12𝑥×2)+(36×𝑥)−(36×2)=𝑥−2𝑥+12𝑥−24𝑥+36𝑥−72.

Finally, we combine like terms: 𝑥−2𝑥+12𝑥−24𝑥+36𝑥−72=𝑥+(−2+12)𝑥+(−24+36)𝑥−72=𝑥+10𝑥+12𝑥−72.

Hence, (𝑥−2)(𝑥+6 )=𝑥+10𝑥+12𝑥−7 2.

We can use the same process to expand a binomial raised to greater powers than 2. For example, we can expand and simplify (2𝑥−𝑦) by writing it as a product of three binomials: (2𝑥−𝑦)=(2𝑥−𝑦)(2𝑥−𝑦)(2𝑥−𝑦).

We can start with the product of two of the binomials and evaluate this using the horizontal method: (2𝑥−𝑦)(2𝑥 −𝑦)=2𝑥(2𝑥−𝑦)− 𝑦(2𝑥−𝑦)=4𝑥−2𝑥 𝑦−2𝑥𝑦+𝑦.

We can then combine like terms to get 4𝑥−2𝑥𝑦−2𝑥 𝑦+𝑦=4𝑥−4𝑥𝑦+𝑦. 

We can then substitute this into the expanded expression to get (2𝑥−𝑦)=(2𝑥−𝑦)4𝑥−4𝑥𝑦+𝑦.

We now need to distribute the multiplication of the factor (2𝑥−𝑦) over the addition and subtraction in the trinomial:

Each term on the right-hand side of this equation is the product of a monomial and a binomial. We can now simplify each term by distributing the monomial over the binomial and then by using the product rule for exponents.

In the first term, we have 4𝑥(2𝑥−𝑦)= 4𝑥×2𝑥−4𝑥×𝑦 =8𝑥−4𝑥𝑦. 

In the second term, we have −4𝑥𝑦(2𝑥−𝑦)=(−4𝑥𝑦×2𝑥)+((−4𝑥𝑦)×(−𝑦))=−8𝑥𝑦+4𝑥𝑦.

In the third term, we have 𝑦(2𝑥− 𝑦)=2𝑥𝑦−𝑦.

The sum of these three expressions then gives us the expanded product of the binomial and trinomial in the question: (2𝑥−𝑦 )=4𝑥(2𝑥−𝑦)−4𝑥 𝑦(2𝑥−𝑦)+𝑦(2𝑥− 𝑦)=8𝑥−4𝑥𝑦−8𝑥𝑦 +4𝑥𝑦+2𝑥𝑦−𝑦. 

We can finally simplify by combining like terms: 8𝑥−4𝑥𝑦−8𝑥𝑦+4 𝑥𝑦+2𝑥𝑦−𝑦=8𝑥+( −4−8)𝑥𝑦+(4+2)𝑥𝑦−𝑦=8𝑥−12𝑥𝑦+6𝑥𝑦−𝑦. 

Therefore, we have shown that (2𝑥−𝑦)=8𝑥−12𝑥 𝑦+6𝑥𝑦−𝑦. 

In the previous example, we saw that we can use this process to expand the product of three binomial factors such that two are the same. In our next example, we will apply this process to find the product of three binomial factors.

Example 4: Expanding the Product of Three Binomials

Expand the expression (5𝑥+2𝑦)(𝑦 −2𝑥), giving your answer in its simplest form.

Answer

We want to expand and simplify this expression. We can do this by first expanding the square of the binomial:

Each term is now the product of a monomial and a binomial. We can expand and simplify each term by distributing the monomial over the binomial and using the product rule for exponents, which states that for any positive integers 𝑚 and 𝑛, we have 𝑥× 𝑥=𝑥.

For the first term, we have

In the second term, we have

The sum of these two expressions then gives us the expanded square of the binomial: 𝑦(𝑦−2𝑥)−2𝑥(𝑦−2𝑥)=𝑦−2𝑥𝑦−2𝑥𝑦+4𝑥=𝑦−4𝑥𝑦+4𝑥. 

We can substitute this into the original expression to obtain (5𝑥+2 𝑦)(𝑦−2𝑥)=(5𝑥+ 2𝑦)𝑦−4𝑥𝑦+4𝑥 .

We can now follow the same process to distribute the factor of 5𝑥+2𝑦 over the trinomial. First, we distribute the binomial over every term in the trinomial: (5𝑥+2𝑦)𝑦−4𝑥𝑦+4𝑥=𝑦(5𝑥+2𝑦)−4𝑥𝑦(5𝑥+2𝑦)+4𝑥(5𝑥+2𝑦).

Second, we distribute each monomial over the binomials: 𝑦(5𝑥+2𝑦)−4𝑥𝑦(5𝑥+2𝑦)+ 4𝑥(5𝑥+2𝑦)=𝑦 ×5𝑥+𝑦×2𝑦+([−4𝑥𝑦×5𝑥]+[−4𝑥𝑦×2𝑦])+4𝑥 ×5𝑥+4𝑥×2𝑦. 

We can now simplify each term using the commutativity of multiplication and the product rule for exponents: 𝑦×5𝑥+𝑦×2 𝑦+([−4𝑥𝑦×5𝑥] +[−4𝑥𝑦×2𝑦])+ 4𝑥×5𝑥+4𝑥×2𝑦=5𝑥𝑦+2𝑦−20𝑥𝑦−8𝑥𝑦+20𝑥+8𝑥𝑦. 

Finally, we rearrange and combine like terms to get 5𝑥𝑦+2𝑦−20 𝑥𝑦−8𝑥𝑦+20𝑥+8𝑥 𝑦=(5−8)𝑥𝑦+2𝑦+ (−20+8)𝑥𝑦+20𝑥=−3𝑥𝑦+2𝑦−12𝑥𝑦+20𝑥=20𝑥−12𝑥𝑦−3𝑥𝑦+2𝑦. 

Hence, (5𝑥+2𝑦)(𝑦−2𝑥)=20𝑥−12𝑥𝑦−3𝑥𝑦+2𝑦.

In our next example, we will find the value of an unknown by expanding a product of polynomials.

Example 5: Finding an Unknown by Expanding a Product of Polynomials

If (𝑥+𝑘)𝑥−4𝑥+9=𝑥−11𝑥+37𝑥−63, find the value of parameter 𝑘.

Answer

In order to determine the value of the unknown parameter 𝑘, we want both sides of the equation to be in the same form. We can do this by expanding the product on the left-hand side of the equation and simplifying.

We can distribute the binomial over each term in the trinomial to get

We can then distribute each product to obtain 𝑥(𝑥+𝑘)−4𝑥(𝑥+𝑘)+9(𝑥+𝑘)=𝑥+𝑘𝑥−4𝑥−4𝑘𝑥+9𝑥+9𝑘.

We can now combine like terms to get 𝑥+𝑘𝑥−4𝑥−4𝑘𝑥+9𝑥+9𝑘=𝑥+(𝑘−4)𝑥+(9−4𝑘)𝑥+9𝑘.

This expression must be equivalent to the right-hand side of the given equation. Therefore, we have 𝑥+(𝑘−4)𝑥+(9−4𝑘)𝑥+9𝑘=𝑥−11𝑥+37𝑥−63.

Both sides of the equations are cubic polynomials in 𝑥, so for them to be equal, their coefficients must be equal. We can equate any coefficients to find 𝑘. We will set the constants on both sides of the equation to be equal: 9𝑘=−63.

Dividing through by 9 gives us 𝑘=−639=− 7.

We can verify that this is correct by substituting 𝑘=−7 into the left-hand side of the equation and simplifying: 𝑥+(𝑘−4)𝑥+(9−4𝑘)𝑥+9𝑘=𝑥+(−7−4)𝑥+(9−4(−7))𝑥+9(−7)=𝑥−11𝑥+(9+28)𝑥−63=𝑥−11𝑥+37𝑥−63.

This is equal to the right-hand side of the given equation.

Hence, 𝑘=−7.

In our final example, we will use this process of distributing a binomial over a trinomial to find an expression for the volume of a rectangular prism.

Example 6: Finding an Expression for the Volume of a Rectangular Prism by Multiplying a Binomial by a Trinomial

A rectangular prism has its base area given by the trinomial (2 𝑥+7𝑥𝑦+2𝑦) cm2 and its height given by the binomial (𝑥+𝑦) cm. Find its volume.

Answer

We first recall that the volume of any prism is given by the product of the area of its base and its height. We are given both of these as expressions. Thus, we can find an expression for the volume in cubic centimeters: volum ecm=(𝑥+𝑦)2𝑥 +7𝑥𝑦+2𝑦.

We want to expand and simplify this expression. We can do this by distributing the factor of 𝑥+𝑦 over the trinomial:

Each term is now the product of a monomial and a binomial. We can expand and simplify each term by distributing the monomial over the binomial and using the product rule for exponents, which states that for any positive integers 𝑚 and 𝑛, we have 𝑥× 𝑥=𝑥.

For the first term, we have

In the second term, we have

In the third term, we have

The sum of these three expressions then gives us the expanded product (𝑥+𝑦)2𝑥+7𝑥𝑦+2𝑦=2𝑥+2𝑥𝑦+7𝑥𝑦+7𝑥𝑦+2𝑥𝑦+2𝑦.

We can simplify by combining like terms: 2𝑥+2𝑥𝑦+7𝑥 𝑦+7𝑥𝑦+2𝑥𝑦+2𝑦= 2𝑥+(2+7)𝑥𝑦+(7+2)𝑥𝑦+2𝑦=2𝑥+9𝑥𝑦+9𝑥𝑦+2𝑦. 

Hence, the volume of the rectangular prism is given by 2𝑥+9𝑥𝑦+9 𝑥𝑦+2𝑦 cm3.

Let’s finish by recapping some of the important points from this explainer.

Key Points

• We can distribute a binomial over an algebraic expression by using the distributive property of multiplication over addition and subtraction. This means multiplying every term in one set of parentheses by every term in the other pair of parentheses and adding the results. We can then simplify by using the product rule for exponents.
• The area model is a good way to calculate the expansion of the product of algebraic expressions.
• We can use this process to evaluate the cube of a binomial by first evaluating the square of the binomial and then distributing the final binomial over the result.
• We can use this process to evaluate the product of three binomials by first evaluating the product of two of the binomials and then distributing the final binomial over the result.