Recall that a monomial is a number, a variable, or the product of a number and one or more variables with whole-number exponents. A polynomial is a monomial or a sum of monomials. A polynomial function is a function of the form

where a ≠ 0, the exponents are all whole numbers, and the coefficients are all real numbers. For this function, an is the leading coefficient, n is the degree, and a0 is the constant term. A polynomial function is in standard form when its terms are written in descending order of exponents from left to right.

The end behavior of a function’s graph is the behavior of the graph as x approaches positive infinity (+∞) or negative infinity (−∞). For the graph of a polynomial function, the end behavior is determined by the function’s degree and the sign of its leading coefficient.

To graph a polynomial function, first plot points to determine the shape of the graph’s middle portion. Then connect the points with a smooth continuous curve and sketch the graph using what you know about end behavior.

EXAMPLE:
Graph f(x) = −x³ + x² + 3x − 3

SOLUTION:
To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior

The degree is odd and the leading coefficient is negative. So, f(x) → +∞ as x → −∞ and f(x) → −∞ as x → +∞.

Operations with Polynomials:

The set of integers is closed under addition and subtraction because every sum or difference results in an integer. To add or subtract polynomials, you add or subtract the coeffi cients of like terms. Because adding or subtracting polynomials results in a polynomial, the set of polynomials is closed under addition and subtraction.

EXAMPLE:

Add.

1. 3x³ + 2x² − x − 7 and x³ − 10x² + 8 in a vertical format.

2. 9y³ + 3y² − 2y + 1 and −5y² + y − 4 in a horizontal format.

SOLUTION:

To subtract one polynomial from another, add the opposite. To do this, change the sign of each term of the subtracted polynomial and then add the resulting like terms.

EXAMPLE:

Subtract.

1. 2x³ + 6x² − x + 1 from 8x³ − 3x² − 2x + 9 in a vertical format

2. 3z² + z − 4 from 2z² + 3z in a horizontal format.

SOLUTION:

To multiply two polynomials, you multiply each term of the first polynomial by each term of the second polynomial. As with addition and subtraction, the set of polynomials is closed under multiplication.

EXAMPLE:
Multiply

1. −x² + 2x + 4 and x − 3 in a vertical format.

2. y + 5 and 3y² − 2y + 2 in a horizontal format.

SOLUTION:

Some binomial products occur so frequently that it is worth memorizing their patterns. You can verify these polynomial identities by multiplying.

Consider the expansion of the binomial (a + b)ⁿ for whole number values of n. When you arrange the coefficients of the variables in the expansion of (a + b)ⁿ, you will see a special pattern called Pascal’s Triangle

EXAMPLE:
Use Pascal’s Triangle to expand (x − 2)⁵

SOLUTION:

The coefficients from the fifth row of Pascal’s Triangle are 1, 5, 10, 10, 5, and 1.

When you divide a polynomial f(x) by a nonzero polynomial divisor d(x), you get a quotient polynomial q(x) and a remainder polynomial

The degree of the remainder must be less than the degree of the divisor. When the remainder is 0, the divisor divides evenly into the dividend. Also, the degree of the divisor is less than or equal to the degree of the dividend f(x). One way to divide polynomials is called polynomial long division.

EXAMPLE:
Divide 2x⁴ + 3x³ + 5x − 1 by x² + 3x + 2.

SOLUTION:

Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x² in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

You can check the result of a division problem by multiplying the quotient by the divisor and adding the remainder. The result should be the dividend.

There is a shortcut for dividing polynomials by binomials of the form x − k. This shortcut is called synthetic division.

EXAMPLE:

Divide −x³ + 4x² + 9 by x − 3.

SOLUTION:
Write the coefficients of the dividend in order of descending exponents. Include a “0” for the missing x-term. Because the divisor is x − 3, use k = 3. Write the k-value to the left of the vertical bar.

Bring down the leading coefficient. Multiply the leading coefficient by the k-value. Write the product under the second coefficient. Add.

Multiply the previous sum by the k-value. Write the product under the third coefficient. Add. Repeat this process for the remaining coefficient. The fi rst three numbers in the bottom row are the coefficients of the quotient, and the last number is the remainder.

The Remainder Theorem tells you that synthetic division can be used to evaluate a polynomial function. So, to evaluate f(x) when x = k, divide f(x) by x − k. The remainder will be f(k).

EXAMPLE:

Use synthetic division to evaluate f(x) = 5x³ − x² + 13x + 29 when x = −4.

SOLUTION:

Previously, you factored quadratic polynomials. You can also factor polynomials with a degree greater than 2. Some of these polynomials can be factored completely using techniques you have previously learned. A factorable polynomial with integer coefficients is factored completely when it is written as a product of unfactorable polynomials with integer coefficients.