Polynomials: Roots, Long Division, and Synthetic Division

Polynomial Functions

Definition and Classification

Polynomial functions are algebraic expressions consisting of variables and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents. They are classified based on the number of terms:

• Monomial: One term (e.g., $3x^4$)

• Binomial: Two terms (e.g., $2x + 7$)

• Trinomial: Three terms (e.g., $5x^2 + 2y - 4$)

Each term consists of a coefficient (numerical factor), variable, and exponent. The constant term has no variable.

Example:

• $5x^2 + 2y - 7$ is a trinomial with coefficients 5, 2, and -7.

Roots of Polynomial Functions

Definition and Multiplicity

The roots (or zeros) of a polynomial function $f(x)$ are the values of $x$ for which $f(x) = 0$. If $(x - a)^k$ is a factor of $f(x)$, then $x = a$ is a root with multiplicity $k$.

• If $k$ is odd, the graph crosses the x-axis at $x = a$.

• If $k$ is even, the graph touches but does not cross the x-axis at $x = a$.

Example:

• For $f(x) = (x - 2)^2(x + 1)(x - 3)^4(x + 2)^3$:

  • Root $x = 2$ has multiplicity 2 (graph touches x-axis).

  • Root $x = -1$ has multiplicity 1 (graph crosses x-axis).

  • Root $x = 3$ has multiplicity 4 (graph touches x-axis).

  • Root $x = -2$ has multiplicity 3 (graph crosses x-axis).

Key Point:

The number of real roots of a polynomial function is equal to the number of times the graph crosses or touches the x-axis, counting multiplicities.

Long Division of Polynomials

Division Algorithm

Long division is a systematic process for dividing polynomials, similar to numerical long division. For polynomials $P(x)$ (dividend) and $D(x)$ (divisor), there exist unique polynomials $Q(x)$ (quotient) and $R(x)$ (remainder) such that:

$P(x) = D(x)Q(x) + R(x)$, where $ext{deg}(R(x)) < ext{deg}(D(x))$

Steps for Long Division:

1. Divide the leading term of the dividend by the leading term of the divisor.

2. Multiply the entire divisor by the result and subtract from the dividend.

3. Repeat the process with the new polynomial (remainder) until the degree of the remainder is less than the degree of the divisor.

Example:

• Divide $2x^3 + 3x^2 - 5x + 7$ by $x - 2$:

• Result: $2x^2 + 7x + 9$ with remainder $25$

Synthetic Division

Definition and Process

Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form $x - c$. It uses only the coefficients and is more efficient than long division for this case.

Steps for Synthetic Division:

1. Write the coefficients of the dividend in order.

2. Write the zero of the divisor ($c$ for $x - c$) to the left.

3. Bring down the leading coefficient.

4. Multiply by $c$ and add to the next coefficient; repeat for all coefficients.

5. The final number is the remainder; the other numbers are coefficients of the quotient.

Example:

• Divide $2x^3 + 3x^2 - 5x + 7$ by $x - 2$ using synthetic division:

• Coefficients: 2, 3, -5, 7; $c = 2$

• Result: Quotient coefficients $2, 7, 9$; remainder $25$

Comparison Table: Long Division vs. Synthetic Division

Method	Divisor Form	Process	When to Use
Long Division	Any polynomial	Uses all terms	General cases
Synthetic Division	Linear ($x - c$)	Uses coefficients only	Quick division by linear factors

Summary

• Polynomials are classified by the number of terms and degree.

• Roots of polynomials can have multiplicity, affecting how the graph interacts with the x-axis.

• Long division and synthetic division are essential tools for dividing polynomials and finding factors.

Additional info: Synthetic division is only applicable for divisors of the form $x - c$. For higher-degree divisors, use long division.