A polynomial is an expression with powers of x, such as 3x² − 5x + 2. In this course we learn the words for its parts, how to find its value, which numbers make it zero, and what the remainder is when we divide it.
3 lessons, about 1 h 15 min3 simulations10 problems6 practice questionsPython in every lesson
Created by StemPy LabA polynomial in one variable x is a sum of terms, each a constant coefficient times a whole number power of x. The highest power with a non-zero coefficient is the degree, and polynomials of degree 0, 1, 2 and 3 are called constant, linear, quadratic and cubic. Powers such as x to the half or x to the minus one are not allowed. Monomials, binomials and trinomials have one, two and three terms. The value of a polynomial at a number is found by putting that number for x. The sum, difference and product of polynomials are polynomials, and the degree of a product is the sum of the degrees.
A number a is a zero of the polynomial p(x) if p(a) is 0. The zero of the linear polynomial ax plus b is minus b over a. A non-zero constant has no zero, a linear polynomial has exactly one, and a polynomial of degree n has at most n zeroes. Geometrically the zeroes are the x coordinates of the points where the graph of y equals p(x) meets the x axis. We can find whole number zeroes by trying values, and some polynomials, such as x squared plus 1, have no real zero.
When a polynomial p(x) is divided by a polynomial, p(x) equals the divisor times the quotient plus the remainder, and the remainder has a smaller degree than the divisor. The remainder theorem says that when p(x) is divided by x minus a, the remainder is p(a), which is found without dividing. For a divisor ax plus b, use the number minus b over a. Synthetic division gives the quotient and the remainder quickly. If the remainder is 0, then a is a zero of the polynomial and x minus a divides it exactly.
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