Use synthetic division to find the quotient and remainder calculator

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Before we can explain how to divide polynomials using synthetic division, let's refresh a few basic notions:

What are polynomials?

A polynomial is an expression involving a sum of non-negative integer powers of at least one variable, each multiplied by real (or complex) numbers, which we call coefficients.

A polynomial in one variable, x (a univariate polynomial), is given by

anxn + an-1xn-1 + ... + a1x + a0,

where an, an-1,..., a1, a0 are the coefficients. We call the individual terms of the form akxk monomials. The leading coefficient of this polynomial is the coefficient of the term with the highest power of x, i.e., the coefficient an, provided that an ≠ 0. We say a polynomial is monic if its leading coefficient is equal to one: an = 1.

The degree of a polynomial is the value of the greatest exponent present in the polynomial with a non-zero coefficient. The polynomial written above has degree n, provided that an ≠ 0. Constant non-null polynomials have degree zero. A null polynomial has its degree left undefined or, sometimes, defined as -∞ (negative infinity). We usually denote the degree of a polynomial with deg.

Polynomial division

The division of polynomials is analogous to dividing integers with remainder, which you've most probably encountered in arithmetic. Let P(x) and D(x) be two polynomials. If D(x) is non-zero, then there exist two polynomials, Q(x) and R(x), which satisfy:

P(x) = D(x) ⋅ Q(x) + R(x)

and deg(R) < deg(D). Moreover, Q(x) and R(x) are unique, i.e., there's no other pair of polynomials that satisfy these two conditions.

The terms we use in polynomial division are analogous to those in arithmetic: P(x) is called the dividend, D(x) is the divisor, Q(x) is a quotient, and R(x) is the remainder.

Note that:

• R(x) = 0 if, and only if, P(x) has D(x) as a factor; and
• If deg(P) < deg(Q), then D(x) = 0 and P(x) = R(x).

The standard way of calculating the quotient and remainder, given a dividend and divisor, is via the algorithm called the polynomial long division.

What is the synthetic division of polynomials?

Synthetic division is a shortcut way of dividing polynomials. It gives the same results as the polynomial long division but is much faster as it involves only the coefficients of the dividend and divisor, on which we perform basic arithmetic operations. As a result, we obtain the coefficients of the quotient and the remainder.

At a first look, you may find synthetic division a bit complicated, but rest assured: once you get the hang of it, you'll never look back!

Synthetic division is most commonly used when dividing by linear monic polynomials x - b. Dividing by such polynomials is very important in the context of finding zeroes and factoring polynomials: to verify whether b is a root of a polynomial, we can synthetically divide this polynomial by x - b and check if the remainder is equal to zero. For details, check out the section below, where we discuss how to use synthetic division to find the zeros of a polynomial.

Keep in mind that synthetic division works for any polynomial divisors: for non-monic polynomials as well as for polynomials of degrees higher than one. However, it becomes more and more complicated as the degree of the divisor grows. In this article, we'll discuss in detail some synthetic division examples of non-monic linear polynomials b1x + b0 and quadratic polynomials c2x2 + c1x + c0.

So, let's dive in and learn how to divide polynomials using synthetic division!

What is synthetic division calculator?