Polynomials Complete - Class 10 Telangana Board Mathematics

Polynomials: Definitions, Properties, and Graphs

Algebraic Expressions, Terms, Constants, and Variables

An algebraic expression is a combination of variables (letters like $x$, $y$, $z$), constants (fixed numbers like $1$, $2$, $3$), and mathematical operations (addition, subtraction, multiplication, division). A term is formed by the product of constants and variables. Variables can change their value, while constants remain fixed.

• Variable: A symbol whose value can change (e.g., $x$, $y$, $a$, $b$).
• Constant: A fixed numerical value (e.g., $1$, $2$, $10$).
• Term: Combination of constants and variables (e.g., $3x$, $4y$).
• Algebraic Expression: A sum or difference…

Polynomials: Definitions, Properties, and Graphs

Algebraic Expressions, Terms, Constants, and Variables

An algebraic expression is a combination of variables (letters like $x$, $y$, $z$), constants (fixed numbers like $1$, $2$, $3$), and mathematical operations (addition, subtraction, multiplication, division). A term is formed by the product of constants and variables. Variables can change their value, while constants remain fixed.

• Variable: A symbol whose value can change (e.g., $x$, $y$, $a$, $b$).
• Constant: A fixed numerical value (e.g., $1$, $2$, $10$).
• Term: Combination of constants and variables (e.g., $3x$, $4y$).
• Algebraic Expression: A sum or difference of terms (e.g., $3x + 4y - 5$).

Definition and Identification of Polynomials

A polynomial is a specific type of algebraic expression where the powers of the variable(s) are non-negative whole numbers. For an expression to be a polynomial, all exponents of variables must be whole numbers (i.e., $0, 1, 2, \ldots$), and no variable should have a negative or fractional exponent.

• Polynomial: An algebraic expression with variables having only non-negative integer exponents.
• Examples of Polynomials: $3x^2 + 4x + 3$, $2x^4 + 5x^2 + 1$.
• Not Polynomials: $3/x + 2x^{100} + 5$ (since $x$ in denominator implies negative exponent), $1/(2x) + 3$.

Structure of a Polynomial: Coefficient, Degree, and Terms

A polynomial consists of terms, each with a coefficient (the numerical part), variable(s), and exponent(s). The degree of a polynomial is the highest power of the variable present in the expression.

• Coefficient: The numerical factor of a term (e.g., in $3y$, $3$ is the coefficient of $y$).
• Degree: The highest exponent of the variable in the polynomial.
• Example: In $x^{99} + 6x^2 + 4/x + 4$, check if all terms are polynomials before determining the degree.

Types of Polynomials Based on Number of Terms

Polynomials are classified based on the number of terms they contain:

• Monomial: A polynomial with only one term (e.g., $5x$, $7$, $x^3$).
• Binomial: A polynomial with two terms (e.g., $x + 5$, $3x - 4$).
• Trinomial: A polynomial with three terms (e.g., $ax^2 + bx + c$).
• Multinomial: A polynomial with more than three terms.

Types of Polynomials Based on Degree

Polynomials are also classified according to their degree:

• Constant Polynomial: Degree $0$ (e.g., $5$, $1000$).
• Linear Polynomial: Degree $1$ (e.g., $ax + b$).
• Quadratic Polynomial: Degree $2$ (e.g., $ax^2 + bx + c$).
• Cubic Polynomial: Degree $3$ (e.g., $ax^3 + bx^2 + cx + d$).
• Zero Polynomial: All coefficients are zero; its degree is undefined.

Zeros/Roots of a Polynomial

The zeros (or roots) of a polynomial are the values of $x$ for which the polynomial equals zero. For a polynomial $p(x)$, if $p(\alpha) = 0$, then $\alpha$ is called a zero or root of the polynomial.

• Linear Polynomial ($ax + b$): At most one root, given by $x = -b/a$.
• Quadratic Polynomial ($ax^2 + bx + c$): At most two roots.
• Cubic Polynomial ($ax^3 + bx^2 + cx + d$): At most three roots.
• The number of roots is at most equal to the degree of the polynomial.

Relationship Between Roots and Coefficients

For quadratic and cubic polynomials, there are explicit relationships between the roots and the coefficients.

• Quadratic ($ax^2 + bx + c$):
  • Sum of roots ($\alpha + \beta$): $-\frac{b}{a}$
  • Product of roots ($\alpha\beta$): $\frac{c}{a}$
• Cubic ($ax^3 + bx^2 + cx + d$):
  • Sum of roots ($\alpha + \beta + \gamma$): $-\frac{b}{a}$
  • Sum of product of roots taken two at a time ($\alpha\beta + \beta\gamma + \gamma\alpha$): $\frac{c}{a}$
  • Product of roots ($\alpha\beta\gamma$): $-\frac{d}{a}$

Value of a Polynomial at a Point

The value of a polynomial at a specific value of $x$ is found by substituting that value into the polynomial.

• For $p(x) = 2x^2 + 3x + 6$, at $x = 1$:
  • $p(1) = 2(1)^2 + 3(1) + 6 = 2 + 3 + 6 = 11$
• The result is called the value of the polynomial at $x = 1$.

Division Algorithm for Polynomials

The division algorithm for polynomials states that for polynomials $p(x)$ (dividend), $g(x)$ (divisor), there exist polynomials $q(x)$ (quotient) and $r(x)$ (remainder) such that:

• The degree of $r(x)$ is less than the degree of $g(x)$.
• Example: For $p(x) = x^2 + 2x + 3$ and $g(x) = x + 1$, perform division to find $q(x)$ and $r(x)$.

Graphs of Polynomials and Finding Zeros

The graph of a polynomial helps visualize its zeros (where the graph intersects the $x$-axis). For quadratic polynomials, the graph is a parabola.

• Linear Polynomial: Graph is a straight line; one intersection with $x$-axis.
• Quadratic Polynomial: Graph is a parabola; can intersect $x$-axis at two points (distinct real roots), one point (equal roots), or not at all (no real roots).
• Cubic Polynomial: Graph is S-shaped or curve-shaped; can have up to three real roots.

Procedure to Draw the Graph and Find Zeros:

1. Substitute various $x$ values into $p(x)$ to get corresponding $y$ values.
2. Tabulate $x$ and $y$ values.
3. Plot the points $(x, y)$ on the coordinate plane.
4. Join the points smoothly to form the graph.
5. The $x$-coordinates where the graph crosses the $x$-axis are the zeros of the polynomial.

Example Table for $p(x) = x^2 + 2x - 3$:

• Zeros are at $x = -3$ and $x = 1$.

Special Cases: Zero and Constant Polynomials

• Zero Polynomial: All coefficients are zero; degree is undefined.

• Constant Polynomial: Only a constant term, degree is zero.

• For zero polynomial, any power of $x$ multiplied by zero remains zero.

• For constant polynomial, $x^0 = 1$, so the degree is zero.

Common Mistakes and Exam Pitfalls

Students often make mistakes by not checking if an expression is a polynomial before finding its degree. Always verify that all exponents are non-negative integers.

• Do not assign a degree to an expression unless it is confirmed to be a polynomial.
• Negative or fractional exponents disqualify an expression from being a polynomial.
• When using the division algorithm, ensure the degree of the remainder is less than the degree of the divisor.
• In graph-based questions, missing a value in the table can lead to incorrect graphs and missed zeros.

Practice and Exam Focus

• Practice finding zeros, drawing graphs, and applying the division algorithm.
• For graph questions, always mention the scale on both axes.
• For value-based questions, substitute the given $x$ values carefully.
• For constructing polynomials from given roots or sum/product of roots, use the relationships stated above.

Key Formulas (as stated/derived):

• Linear root: $x = -b/a$
• Quadratic sum of roots: $-\frac{b}{a}$, product: $\frac{c}{a}$
• Cubic sum of roots: $-\frac{b}{a}$, sum of products two at a time: $\frac{c}{a}$, product: $-\frac{d}{a}$
• Division algorithm: $p(x) = g(x)q(x) + r(x)$

Focus on these concepts and methods for maximum marks in polynomial-related questions.