\displaystyle \left(w + x\right)\left(y + z\right)	\displaystyle =	\displaystyle \left(w + x\right)y + \left(w + x\right)z	Distributive property once
	\displaystyle =	\displaystyle y\left(w + x\right) + z\left(w + x\right)	Reorder variables
	\displaystyle =	\displaystyle yw + yx + zw + zx	Distributive property twice

In general, we can multiply any two polynomials using this process. Here is an example of a trinomial multiplied by a binomial:

\displaystyle \left(x^2 + 3x + 1\right)\left(x - 2\right)	\displaystyle =	\displaystyle x\left(x^2 + 3x + 1\right) - 2\left(x^2 + 3x + 1\right)	Distributive property once
	\displaystyle =	\displaystyle x^3 + 3x^2 + x - 2x^2 - 6x - 2	Distributive property twice
	\displaystyle =	\displaystyle x^3 + x^2 - 5x - 2	Combine like terms

Notice that we can summarize the process of distributing twice as "multiply each term in the first polynomial by each term in the second polynomial, and add the results". We then simplify by combining like terms, if possible.

Worked examples

Example 1

Multiply 3 x \left( 2 x^{2} - 5 x + 4\right).

Solution

We can use the distributive property to get the product of the monomial 3 x and the trinomial 2 x^{2} - 5 x + 4.

\displaystyle 3 x \left( 2 x^{2} - 5 x + 4\right)

\displaystyle =

\displaystyle 6 x^{3} - 15 x^{2} + 12x

Since there are no more like terms and the expression is already in standard form, the final answer is 6 x^{3} - 15 x^{2} + 12x.

Example 2

Consider the polynomials 7 y + 2 and 4 y - 5.

Find the product of the two polynomials.

Approach

Since we want to find the product of the two polynomials, we want to mutliply them together.

Solution

We can use the distributive property to get the product of the two binomials 7y + 2 and 4 y - 5.

\displaystyle \left( 7 y + 2\right) \left( 4 y - 5\right)	\displaystyle =	\displaystyle 4y\left( 7 y + 2\right) - 5\left( 7 y + 2\right)	Distribute \left( 7 y + 2\right)
	\displaystyle =	\displaystyle 28 y^{2} + 8y - 35 y - 10	Distribute 4y and - 5
	\displaystyle =	\displaystyle 28y^{2} - 27y - 10	Combine like terms

Since the expression is already in standard form, the final answer is 28y^{2} - 27y - 10.

Explain why the product of two polynomials is also a polynomial.

Solution

A polynomial is a collection of terms in the form mx^n where m is a real number and n is a non-negative integer.

When multiplying polynomials together, the exponents of terms with the same base are added together. The result of adding two non-negative integers must also be a non-negative integer, so the exponents of the product's terms will be non-negative integers.

Since polynomials are a sum of terms with positive integer exponents, or constants, when we multiply these together, our result is always of the form mx^n.

Reflection

We can use this explanation to think about what happens when we perform multiple operations on polynomials. What happens if we add two polynomials and multiply this result by another polynomial? What if we multiply three polynomials? Is our result still a polynomial?

Outcomes

M2.N.Q.A.1

Use units as a way to understand real-world problems.*

M2.N.Q.A.1.B

Use appropriate quantities in formulas, converting units as necessary.

M2.A.APR.A.1

Add, subtract, and multiply polynomials. Use these operations to demonstrate that polynomials form a closed system that adhere to the same properties of operations as the integers.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP2

Reason abstractly and quantitatively.

M2.MP4

Model with mathematics.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

1.02 Multiplying polynomials

Concept summary

Worked examples

Example 1

Solution

Example 2

Approach

Solution

Solution

Reflection

Outcomes

M2.N.Q.A.1

M2.N.Q.A.1.B

M2.A.APR.A.1

M2.MP1

M2.MP2

M2.MP4

M2.MP5

M2.MP6

M2.MP7

M2.MP8

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