Introduction to Polynomials

The One Thing

Polynomials are classified by degree, and their zeros connect directly to factors and remainders. In particular, the Remainder Theorem gives a rapid method for division by a linear polynomial, while the Factor Theorem identifies factors through zeros.

Definitions and Results

• Polynomial: An expression of the form $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$, where the coefficients are real numbers and the powers of $x$ are non-negative integers. A polynomial cannot contain variables with negative powers, fractional powers, or variables in denominators; therefore, $1/x$ and $x^{1/2}$ are not polynomials in $x$.

• Term: Each part of a polynomial separated by a plus or minus sign, such as $3x^2$, $-5x$, or $7$.

• Coefficient: The numerical factor of a term. In $6x^3$, $6$ is the coefficient of $x^3$.

• Constant term: A term without a variable, such as $8$ in $2x^2-3x+8$.

• Degree of a polynomial: The greatest exponent of the variable with a non-zero coefficient. It is determined by the highest power, not by the number of terms.

• Linear polynomial: A polynomial of degree $1$, such as $4x-7$.

• Quadratic polynomial: A polynomial of degree $2$, such as $x^2+5x+6$.

• Cubic polynomial: A polynomial of degree $3$, such as $2x^3-x+4$.

• Monomial, binomial and trinomial: A polynomial with one term is a monomial, one with two terms is a binomial, and one with three terms is a trinomial.

• Zero or root: A number $k$ is a zero of $p(x)$ if $p(k)=0$.

• Value of a polynomial: The result obtained after replacing the variable by a given number.

• Factor: An expression that divides another expression exactly, leaving remainder zero.

• Remainder Theorem: When $p(x)$ is divided by $x-a$, the remainder is $p(a)$. When the divisor is $x+a$, write it as $x-(-a)$; the remainder is therefore $p(-a)$.

• Factor Theorem: For a polynomial $p(x)$, $x-a$ is a factor if and only if $p(a)=0$. Thus, $p(a)=0$, $x-a$ being a factor, and division by $x-a$ having remainder zero are equivalent statements.

• Division algorithm: For polynomials $p(x)$ and $g(x)$, with $g(x)\ne 0$,

$$p(x)=g(x)q(x)+r(x),$$ where the degree of $r(x)$ is less than the degree of $g(x)$, or $r(x)=0$. Here, $p(x)$ is the dividend, $g(x)$ the divisor, $q(x)$ the quotient, and $r(x)$ the remainder. When the divisor is linear, the remainder is a constant.

• Zero polynomial: The polynomial whose coefficients are all zero. Its degree is not defined, and it is neither linear, quadratic, nor cubic. A non-zero constant polynomial has degree $0$.

• Number of zeros: A polynomial of degree $n$ can have at most $n$ distinct zeros.

• Graphical interpretation: The graph of a polynomial crosses or touches the $x$-axis at the real zeros of the polynomial.

Worked Methods

1. Classifying a polynomial

• Identify the highest exponent of the variable with a non-zero coefficient.
• Use this exponent to determine the degree.
• Classify the polynomial:

- Degree $1$: linear. - Degree $2$: quadratic. - Degree $3$: cubic.

• Count the terms if the question asks whether it is a monomial, binomial, or trinomial.

Examples:

• $4x-7$ is a linear polynomial.
• $x^2+5x+6$ is a quadratic polynomial.
• $2x^3-x+4$ is a cubic polynomial.

2. Finding the value of a polynomial

• Substitute the given value for the variable.
• Simplify the resulting numerical expression.

For example, if $$p(x)=x^2-5x+6,$$ then $$p(2)=2^2-5(2)+6=4-10+6=0,$$ and $$p(3)=3^2-5(3)+6=9-15+6=0.$$

Therefore, $2$ and $3$ are zeros of $p(x)$.

3. Applying the Remainder Theorem

To find the remainder when $p(x)$ is divided by $x-a$:

• Identify the value $a$ from the divisor $x-a$.
• Calculate $p(a)$.
• State that $p(a)$ is the remainder.

For division by $x+a$:

• Rewrite the divisor as $x-(-a)$.
• Calculate $p(-a)$.
• State that $p(-a)$ is the remainder.

This method is faster than polynomial long division when the divisor is linear.

4. Applying the Factor Theorem

To test whether $x-a$ is a factor of $p(x)$:

• Calculate $p(a)$.
• If $p(a)=0$, then $x-a$ divides $p(x)$ exactly and is a factor.
• If $p(a)\ne 0$, then $x-a$ is not a factor.

Conversely, if $x-a$ is known to be a factor of $p(x)$, then $a$ is a zero of $p(x)$.

For $$p(x)=x^2-5x+6,$$ we have $$p(2)=0\text{and}p(3)=0.$$ Therefore, $2$ and $3$ are zeros, and $$(x-2)\text{and}(x-3)$$ are factors.

5. Using polynomial division

To divide $p(x)$ by $g(x)$:

• Arrange the terms of the dividend and divisor in descending powers of $x$.
• Divide the leading term of the dividend by the leading term of the divisor to obtain the first term of the quotient.
• Multiply the divisor by this quotient term.
• Subtract the result from the dividend.
• Repeat until the remaining polynomial has degree less than the degree of the divisor, or is zero.
• Express the result using

$$p(x)=g(x)q(x)+r(x).$$

The remainder must have lower degree than the divisor. If the divisor is linear, the remainder is a constant. If division is by $x-a$ and the remainder is zero, the Factor Theorem confirms that $x-a$ is a factor.

Where It Goes Wrong

• Confusing the number of terms with the degree; the degree is determined by the highest power with a non-zero coefficient.
• Treating $1/x$ or $x^{1/2}$ as polynomials in $x$, despite the requirement that powers be non-negative integers and variables must not occur in denominators.
• Forgetting that the zero polynomial has no defined degree, while a non-zero constant polynomial has degree $0$.
• Using $p(a)$ for division by $x+a$ instead of rewriting the divisor as $x-(-a)$ and using $p(-a)$.
• Stating that $a$ is a zero without verifying $p(a)=0$.
• Giving a remainder whose degree is not less than the degree of the divisor, contrary to the division algorithm.

What Gets Asked

• Define a polynomial, term, coefficient, constant term, degree, zero, factor, or remainder.
• Classify an expression as linear, quadratic, cubic, monomial, binomial, or trinomial.
• Determine whether an expression is a polynomial in $x$.
• Find the value $p(a)$ after substituting a given number.
• Find the remainder when a polynomial is divided by $x-a$ or $x+a$.
• Test whether $x-a$ is a factor using the Factor Theorem.
• Identify zeros and factors from $p(a)=0$, including the example $p(x)=x^2-5x+6$.
• Perform polynomial division and state the result in the form

$$p(x)=g(x)q(x)+r(x).$$

• Explain the relationship between zeros, factors, and remainders.
• State the maximum number of distinct zeros of a polynomial of degree $n$.
• Interpret real zeros as points where the graph crosses or touches the $x$-axis.