06/14 6. Polynomial Expressions and Functions

Polynomial Structure and Operations

A polynomial expression is a sum of terms of the form $ax^n$, where $a$ is a constant and $n$ is a nonnegative integer. A polynomial function can be written as

where $a_n\neq 0$. The degree of a polynomial is the greatest exponent with a nonzero coefficient. The coefficient of that highest-degree term is the leading coefficient, and $a_0$ is the constant term.

For example, in

the degree is $5$, the leading coefficient is $4$, and the constant term is $7$.

Combining and multiplying expressions

To add or subtract polynomials, combine like terms: terms with the same variable raised to the same power. When subtracting, distribute the negative sign first.

To multiply, use the distributive property so that every term in one factor is multiplied by every term in the other factor:

The degree of a product is the sum of the degrees of its nonzero polynomial factors.

Takeaway: Identify the highest power to describe a polynomial, combine only like terms, and distribute across every term when multiplying.

Polynomial Identities and Factoring

Polynomial identities are equations that hold for every value of the variables. Recognizing them makes factoring faster and reduces the amount of expansion required.

Frequently used identities

For example, $9x^2-25$ is a difference of squares:

When factoring, first check for a greatest common factor. Then look for a familiar identity, such as a difference of squares or a sum or difference of cubes.

Takeaway: Factoring is often the reverse of multiplication, so matching an expression to a standard identity can reveal its factors immediately.

Polynomial Division

Polynomial division follows the same structure as integer division. If a polynomial $f(x)$ is divided by a nonzero polynomial $d(x)$, then

where $q(x)$ is the quotient and the degree of $r(x)$ is less than the degree of $d(x)$.

Long division

To divide $2x^3+3x^2-5x+6$ by $x+2$:

1. Divide the leading terms: $2x^3\div x=2x^2$.

2. Multiply: $2x^2(x+2)=2x^3+4x^2$.

3. Subtract to obtain $-x^2$, bring down the next term, and repeat.

4. Continue until the remainder has lower degree than the divisor.

The result is

Thus, the quotient is $2x^2-x-3$, and the remainder is $12$.

Synthetic division

Synthetic division is a shorter method when the divisor is $x-c$. Since $x+2=x-(-2)$, use $-2$ with the coefficients $2,3,-5,6$:

The last entry is the remainder, and the preceding entries are the coefficients of the quotient. Therefore, the quotient is $2x^2-x-3$, with remainder $12$.

Takeaway: Long division works for general polynomial divisors, while synthetic division provides a compact method for linear divisors.

Remainders and Factors

The Remainder Theorem connects division with evaluation. When $f(x)$ is divided by $x-c$, the remainder is $f(c)$:

For

the remainder after division by $x-3$ is found without performing long division:

Therefore,

for some quadratic polynomial $q(x)$.

The Factor Theorem is a direct consequence:

For example, let

Evaluating at $2$ gives

Thus, $x-2$ is a factor. Synthetic division produces the quotient $x^2-3x-4$, so

Takeaway: Evaluate $f(c)$ to find the remainder for division by $x-c$; a zero remainder proves that $x-c$ is a factor.

Zeros and Multiplicity

A zero or root of $f$ is a number $c$ such that $f(c)=0$. The zeros are the $x$-coordinates of the graph's $x$-intercepts. By the Factor Theorem, each zero $c$ corresponds to a factor $x-c$.

Factoring to find zeros

Suppose

Factoring gives

Set each factor equal to zero:

The zeros are $x=1$, $x=2$, and $x=3$.

Multiplicity and graph behavior

A repeated factor gives a zero with multiplicity greater than one. For

$x=2$ has multiplicity $3$, and $x=-1$ has multiplicity $2$. A zero with odd multiplicity generally causes the graph to cross the $x$-axis, while a zero with even multiplicity generally causes the graph to touch the axis and turn around.

A polynomial of degree $n$ has exactly $n$ complex zeros when multiplicities are counted. Some may be nonreal. For example,

has no real zeros but has the two complex zeros $x=2i$ and $x=-2i$.

Takeaway: Factor the polynomial, set each factor equal to zero, and count repeated factors when checking the total number of complex zeros.

A Strategy for Finding Polynomial Zeros

The Rational Zero Theorem narrows the search for rational zeros of a polynomial with integer coefficients. If

then every rational zero $\frac{p}{q}$, written in lowest terms, must have $p$ as a factor of the constant term $a_0$ and $q$ as a factor of the leading coefficient $a_n$.

For

the possible rational zeros are

A systematic strategy is:

1. Write the polynomial in standard form.

2. Factor out any greatest common factor.

3. Look for identities such as a difference of squares or a sum or difference of cubes.

4. List possible rational zeros.

5. Test candidates by substitution or the Remainder Theorem.

6. Use synthetic division when a zero is found.

7. Factor the reduced polynomial.

8. Set every factor equal to zero and solve.

9. Check the number of zeros against the degree, counting multiplicities.

For the example above, test $x=2$:

Therefore, $x-2$ is a factor. Synthetic division gives

Thus,

The zeros are

Takeaway: The Rational Zero Theorem supplies candidates, the Remainder Theorem tests them, and synthetic division reduces the problem after a candidate succeeds.