5.6: Multiplying Polynomials and the Distributive Property

Media

Videos

• Multiplying Polynomials

Definitions and Theorems

Multiplying polynomials is a direct extension of the distributive property: each term of one factor is multiplied by every term of the other, and the resulting products are combined. This section reviews that process, from a monomial times a polynomial through the product of two higher-order polynomials, along with the special products that arise when a binomial is squared.

The Distributive Property

For all real numbers \(a\), \(b\), and \(c\),\[a(b+c) = ab + ac \qquad \text{and} \qquad (a+b)c = ac + bc.\nonumber\]More generally, a single factor distributes across every term of a sum:\[a(b+c+d) = ab + ac + ad,\nonumber\]and likewise for a sum of any number of terms.

The distributive property is the engine behind every polynomial product. When both factors contain more than one term, you apply it repeatedly—once for each term of the factor being distributed.

Theorem: Product of Two Polynomials

The product of two polynomials is obtained by multiplying each term of the first polynomial by each term of the second and then combining like terms. The result is again a polynomial.

Note: Like Terms

Recall that like terms are terms with the same variables raised to the same powers. Only like terms may be combined, and they are combined by adding their coefficients. After multiplying, collect the like terms and write the result in descending powers of the variable.

Theorem: Square of a Binomial

For all real numbers \(a\) and \(b\),\[(a+b)^2 = a^2 + 2ab + b^2 \qquad \text{and} \qquad (a-b)^2 = a^2 - 2ab + b^2.\nonumber\]In words, the square of a binomial equals the square of the first term, plus twice the product of the two terms, plus the square of the second term.

Caution: Do Not Drop the Middle Term

The square of a binomial is not the sum of the squares:\[(a+b)^2 \neq a^2 + b^2.\nonumber\]The middle term \(2ab\) is essential. Squaring always means multiplying the binomial by itself, so \((a+b)^2 = (a+b)(a+b)\).

Note: Product of a Sum and a Difference

A closely related special product is the product of a sum and a difference of the same two terms:\[(a+b)(a-b) = a^2 - b^2.\nonumber\]The two middle terms cancel, leaving a difference of squares. You will rely on this pattern heavily when factoring.

Caution: FOIL Applies Only to Two Binomials

The mnemonic FOIL (First, Outer, Inner, Last) is only a bookkeeping device for the product of two binomials. It does not extend to a binomial times a trinomial, or to any larger product. For those, return to the general rule: multiply each term of one factor by every term of the other.

Examples

Example \(\PageIndex{1}\): Multiplying a Monomial by a Polynomial

Expand \(-2x^3(4x^2 - 3x - 6)\).

Solution

Distribute the monomial \(-2x^3\) to each term inside the parentheses, multiplying the coefficients and adding the exponents on the common base:\[-2x^3(4x^2) - 2x^3(-3x) - 2x^3(-6).\nonumber\]Evaluating each product, and tracking signs carefully,\[-8x^5 + 6x^4 + 12x^3.\nonumber\]Because there are no like terms, this is the final expanded form.

Example \(\PageIndex{2}\): Multiplying Two Binomials

Expand \((2x+5)(3x-4)\).

Solution

Multiply each term of the first binomial by each term of the second:\[(2x)(3x) + (2x)(-4) + (5)(3x) + (5)(-4).\nonumber\]This gives\[6x^2 - 8x + 15x - 20.\nonumber\]Combining the like terms \(-8x\) and \(15x\),\[6x^2 + 7x - 20.\nonumber\]

Example \(\PageIndex{3}\): A Product in Two Variables

Expand \((x-7y)(x+2y)\).

Solution

Multiply each term of the first factor by each term of the second:\[(x)(x) + (x)(2y) + (-7y)(x) + (-7y)(2y).\nonumber\]This produces\[x^2 + 2xy - 7xy - 14y^2.\nonumber\]The like terms \(2xy\) and \(-7xy\) combine to \(-5xy\), giving\[x^2 - 5xy - 14y^2.\nonumber\]

Example \(\PageIndex{4}\): Squaring a Binomial

Expand each square.

1. \((4x-3)^2\)
2. \((2x+5y)^2\)

Solutions

1. Apply \((a-b)^2 = a^2 - 2ab + b^2\) with \(a=4x\) and \(b=3\):\[(4x)^2 - 2(4x)(3) + 3^2 = 16x^2 - 24x + 9.\nonumber\]
2. Apply \((a+b)^2 = a^2 + 2ab + b^2\) with \(a=2x\) and \(b=5y\):\[(2x)^2 + 2(2x)(5y) + (5y)^2 = 4x^2 + 20xy + 25y^2.\nonumber\]

Example \(\PageIndex{5}\): Product of a Sum and a Difference

Expand \((5x+2)(5x-2)\).

Solution

This is a sum times a difference of the same two terms, so apply \((a+b)(a-b) = a^2 - b^2\) with \(a=5x\) and \(b=2\):\[(5x)^2 - 2^2 = 25x^2 - 4.\nonumber\]As a check, expanding term by term gives \(25x^2 - 10x + 10x - 4\), and the middle terms cancel.

Example \(\PageIndex{6}\): Multiplying Higher-Order Polynomials

Expand each product and write the result in descending powers of \(x\).

1. \((x-3)(x^2-4x+1)\)
2. \((2x^2+x-3)(x^2-4x+1)\)

Solutions

1. Multiply each term of the binomial by every term of the trinomial:\[x(x^2-4x+1) - 3(x^2-4x+1).\nonumber\]Distributing each piece gives\[(x^3 - 4x^2 + x) + (-3x^2 + 12x - 3).\nonumber\]Combining like terms,\[x^3 - 7x^2 + 13x - 3.\nonumber\]
2. Multiply each of the three terms of the first factor by every term of the second:\[2x^2(x^2-4x+1) + x(x^2-4x+1) - 3(x^2-4x+1).\nonumber\]Distributing each piece gives\[(2x^4 - 8x^3 + 2x^2) + (x^3 - 4x^2 + x) + (-3x^2 + 12x - 3).\nonumber\]Collecting like terms by degree,\[2x^4 + (-8+1)x^3 + (2-4-3)x^2 + (1+12)x - 3,\nonumber\]which simplifies to\[2x^4 - 7x^3 - 5x^2 + 13x - 3.\nonumber\]

---

Sources

Several parts of this text use modifications from the following sources:

• Wikipedia article: "Distributive property"
• Wikipedia article: "Polynomial"

All of these sources are released under the Creative Commons Attribution-Share-Alike License 4.0.