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Concept Version 14
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Multiplying Polynomials

To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.

Learning Objective

  • Explain how to multiply polynomials using the distributive property and describe the results of doing so


Key Points

    • To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together.
    • To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.
    • The degree of a product of two polynomials equals the sum of the degrees of said polynomials.
    • The zeros of a product of two polynomial are the zeros of the two factors, combined.

Terms

  • monomial

    An algebraic expression consisting of one term.

  • commutative

    A binary operation is commutative if changing the order of the operands does not change the result, for example addition and multiplication.

  • polynomial

    an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as $a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0$ . Importantly, because all exponents are positive, it is impossible to divide by $x$.


Full Text

Multiplying a polynomial by a monomial is a direct application of the distributive and associative properties. Recall that the distributive property says that

$a(b + c) = ab + ac$

for all real numbers $a,b$ and $c.$ The associative property says that

$(ab)c=a(bc)$

for all real numbers $a,b$ and $c.$ 

As we will treat variables in the same way as real numbers, the same properties hold whenever $a,b$ and/or $c$ is a variable. So for the multiplication of a monomial with a polynomial we get the following procedure:

Multiply every term of the polynomial by the monomial and then add the resulting products together. 

For example,

$\begin{aligned} 3x^2(4x-\pi)&=3x^2 \cdot 4x-3x^2 \cdot \pi \\&=12x^3-3\pi x^2. \end{aligned}$

To multiply a polynomial $P(x) = M_1(x) + M_2(x) + \ldots + M_n(x)$ with a polynomial $Q(x) = N_1(x) + N_2(x) + \ldots + N_k(x)$, where both are written as a sum of monomials of distinct degrees, we get

$\begin{aligned} P(x)Q(x) &= (M_1(x)+M_2(x)+ \ldots + M_n(x))Q(x) \\ &=M_1(x)Q(x) + M_2(x)Q(x) + \ldots + M_n(x)Q(x) \\ &=M_1(x)N_1(x) + \ldots + M_1(x)N_k(x) + \ldots \\ &+M_n(x)N_1(x) + \ldots + M_n(x)N_k(x) \end{aligned}$

and we see that this equals the sum of the products of the terms, where every term of $P(x)$ is multiplied exactly once with every term of $Q(x)$. Notice that since the highest degree term of $P(x)$ is multiplied with the highest degree term of $Q(x)$ we have that the degree of the product equals the sum of the degrees, since

$a^na^m=a^{m+n}$

for all real numbers (and variables) $a$ and all non-negative integers $m$ and $n$.

For convenience, we will use the commutative property of addition to write the expression so that we start with the terms containing $M_1(x)$ and end with the terms containing $M_n(x)$.

This method is commonly called the FOIL method, where we multiply the First, Outside, Inside, and Last pairs in the expression, and then add the products of like terms together.

For example, to find the product of $(2x+3)(x−4)$, use FOIL and then add the products together: 

$\begin {aligned} 2x \cdot x+2x \cdot (−4)+3 \cdot x+3 \cdot(−4)&=2x^2-8x+3x-12 \\ &=2x^2−5x-12 \end {aligned}$

Zeros of a Product of Polynomials

Since we made sure that the product of polynomials abides the same laws as if the variables were real numbers, the evaluation of a product of two polynomials in a given point will be the same as the product of the evaluations of the polynomials:

$P(x_0)Q(x_0) = PQ(x_0)$

for all real numbers $x_0.$ 

In particular $PQ(x_0) = 0$ if and only if $P(x_0)Q(x_0)=0$, if and only if $P(x_0) = 0$ or $Q(x_0) = 0$. So the roots of a product of polynomials are exactly the roots of its factors, i.e. $x_0$ is a zero for $PQ(x)$ if it is a zero for $P(x)$ or for $Q(x)$ (and possibly both).

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