Grade 9 Polynomials

Definition: A polynomial is an algebraic expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Terms: A polynomial is made up of terms. A term is a number, a variable, or a product of numbers and variables with whole number exponents. Terms are separated by + or - signs.
Example: In $3x^2 + 2x - 1$, the terms are $3x^2$, $2x$, and $-1$.

Coefficient: The numerical factor of a term is called its coefficient.
Example: In $3x^2$, the coefficient is 3. In $2x$, the coefficient is 2. In $-1$, the coefficient is -1.

Variable: The letter representing an unknown value (usually x, y, z, etc.).

Constant Term: A term that does not contain a variable (a number).
Example: In $3x^2 + 2x - 1$, the constant term is -1.

Degree of a Term: The sum of the exponents of the variables in a term. For a term with one variable, it's just the exponent of that variable.
Example: The degree of $3x^2$ is 2. The degree of $2x$ (which is $2x^1$) is 1. The degree of a constant term like -1 (which is $-1x^0$) is 0.

Degree of a Polynomial: The highest degree among all the terms in the polynomial.
Example: For $3x^2 + 2x - 1$, the degrees of the terms are 2, 1, and 0. The highest degree is 2, so the degree of the polynomial is 2.

Special Cases:

• The degree of a non-zero constant polynomial (like 5) is 0.
• The degree of the zero polynomial (0) is undefined or sometimes taken as -infinity.

Based on Number of Terms:

• **Monomial:** A polynomial with one term (e.g., $5x^3$).
• **Binomial:** A polynomial with two terms (e.g., $2x + 3$).
• **Trinomial:** A polynomial with three terms (e.g., $x^2 - 4x + 1$).
• Polynomials with more than three terms don't have special names based on the number of terms.

Based on Degree:

• **Linear Polynomial:** A polynomial of degree 1 (e.g., $3x - 5$).
• **Quadratic Polynomial:** A polynomial of degree 2 (e.g., $2x^2 + x - 7$).
• **Cubic Polynomial:** A polynomial of degree 3 (e.g., $x^3 - 2x^2 + 4$).
• Polynomials of higher degrees are referred to by their degree (e.g., a polynomial of degree 4).

Step 1: To evaluate a polynomial $P(x)$ at $x = a$, substitute the value 'a' for every occurrence of 'x' in the polynomial.
Let the polynomial be $P(x) = 2x^2 - 3x + 4$. We want to find $P(2)$.

Step 2: Substitute $x = 2$ into the polynomial.
$P(2) = 2(2)^2 - 3(2) + 4$

Step 3: Simplify the expression using the order of operations (PEMDAS/BODMAS).
$P(2) = 2(4) - 6 + 4$

Step 4: Continue simplifying.
$P(2) = 8 - 6 + 4$

Step 5: Final calculation.
$P(2) = 2 + 4$
$P(2) = 6$

Result: The value of the polynomial $P(x) = 2x^2 - 3x + 4$ at $x = 2$ is 6.

Definition: A real number 'a' is called a zero of a polynomial $P(x)$ if evaluating the polynomial at $x=a$ results in 0, i.e., $P(a) = 0$.

Relationship to Graph: The real zeros of a polynomial are the x-intercepts of its graph (where the graph crosses or touches the x-axis).

Example: Consider the polynomial $P(x) = x - 2$. If we evaluate $P(2)$, we get $P(2) = 2 - 2 = 0$. Since $P(2) = 0$, the number 2 is a zero of the polynomial $P(x) = x - 2$.

Example 2: Consider $Q(x) = x^2 - 4$. If we evaluate $Q(2)$, $Q(2) = 2^2 - 4 = 4 - 4 = 0$. If we evaluate $Q(-2)$, $Q(-2) = (-2)^2 - 4 = 4 - 4 = 0$. The zeros of $Q(x) = x^2 - 4$ are 2 and -2.