Polynomials – Formula & Equation Sheet
Essential formulas and equations from Mathematic, tailored for Class 10 in Mathematics.
This one-pager compiles key formulas and equations from the Polynomials chapter of Mathematic. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Polynomial Degree: The degree of a polynomial p(x) = ax^n + bx^(n-1) + ... + k is n.
p(x) is a polynomial. a (coefficient of highest degree term), n (highest power of x). The degree indicates the highest power in a polynomial, crucial in determining polynomial behavior.
General form of a linear polynomial: p(x) = ax + b.
a (non-zero slope) and b (intercept). Linear polynomials represent straight lines and are foundational in algebra.
General form of a quadratic polynomial: p(x) = ax^2 + bx + c, where a ≠ 0.
a, b, c are constants. Quadratics create parabolas, crucial in various real-world applications like projectile motion.
General form of a cubic polynomial: p(x) = ax^3 + bx^2 + cx + d, where a ≠ 0.
a, b, c, d are constants. Cubics can have one, two, or three real roots and appear in optimization problems.
Sum of the roots of a quadratic: S = -b/a.
S (sum of roots), a (coefficient of x^2), b (coefficient of x). Useful for finding roots without actual solving.
Product of the roots of a quadratic: P = c/a.
P (product of roots), a (coefficient of x^2), c (constant term). Helps in identifying relationships between roots.
Value of a polynomial at x = k: p(k) = ak^2 + bk + c.
k is a specific input. This expression evaluates the polynomial at specified points, essential for graphing.
Zeroes of a polynomial: k is a zero of p(x) if p(k) = 0.
Zeroes are solutions to the polynomial equation. Critical for finding intercepts on graphs.
Factoring a quadratic: p(x) = a(x - r1)(x - r2).
r1, r2 are roots. Useful for solving and graphing quadratic equations in vertex form.
Remainder Theorem: If p(x) is divided by (x - k), then remainder = p(k).
p(x) is the polynomial. Helps in efficiently finding remainders without long division.
Equations
p(x) = x^2 - 3x - 4.
Example of a quadratic polynomial. Can determine roots using factorization or quadratic formula.
p(k) = ak + b, where k is a zero of p(x) = ax + b.
Finding zeros of linear polynomials using their coefficients, crucial in algebraic solutions.
If p(x) = ax^2 + bx + c, then set p(x) = 0 to find roots.
Crucial step in solving quadratic equations, leading to the application of the quadratic formula.
Using Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a).
Calculates the roots of any quadratic equation. Key for algebraic problem-solving.
Division Algorithm: p(x) = (x - k)q(x) + r.
p(x) is divided by (x - k), yielding quotient q(x) and remainder r. Fundamental in polynomial division.
Product of the roots of a quadratic: r1 * r2 = c/a.
Determines the product of the solutions quickly from coefficients, enhancing calculation efficiency.
Evaluating p(0): p(0) = c for p(x) = ax^2 + bx + c.
Finding y-intercept in polynomials, critical for graphing.
Number of turns of a polynomial graph: Maximum of (n-1) for a polynomial of degree n.
Indicates the complexity of the polynomial’s graph. Important in graph sketching.
Symmetrical property of parabolas: For p(x) = ax^2 + bx + c, axis of symmetry is x = -b/(2a).
Helps locate vertex efficiently. Useful in graphing quadratic functions.
Finding coefficients from roots: p(x) = a(x - r1)(x - r2).
Facilitates finding polynomial coefficients through known roots, essential in polynomial construction.
