Algebra Class 10: Fundamental Theorem of Arithmetic

A number that can be expressed as a fraction where the numerator and denominator are both integers, and the denominator is not zero.

A number that cannot be expressed as a fraction of two integers. It's a decimal that goes on forever without repeating.

The smallest positive integer that is a multiple of all the given integers. It's the least common multiple.

The largest positive integer that divides into all the given integers. It's the greatest common factor.

A number that has exactly two factors: 1 and itself.

A number that has more than two factors.

A number that can be written as the product of prime factors.

The process of finding the prime factors of a number and then multiplying them together.

The number of times the graph of a polynomial intersects the x-axis.

The highest power of the variable in a polynomial.

A polynomial of degree 3. It has at most 3 zeroes, meaning it can cross the x-axis at most 3 times.

A polynomial with only one term. It can be a constant or a variable raised to a power.

A polynomial with two terms. It can be a combination of constants and variables with different powers.

A polynomial with three terms. It can be a combination of constants and variables with different powers.

A polynomial with zero as its constant term.

A polynomial with only one variable.

A real number 'k' is a zero of a polynomial, p(x), if plugging 'k' into the polynomial makes the result equal to zero.

The graph of a linear polynomial (ax + b) is a straight line that intersects the x-axis at exactly one point. The x-coordinate of that point represents the zero of the polynomial.

The graph of a quadratic polynomial (ax² + bx + c) is a parabola. It may intersect the x-axis at two points, one point, or not at all. The x-coordinates of these intersection points represent the zeroes of the polynomial.

For a linear polynomial ax + b, the zero is -b / a.

Dividing a polynomial by another polynomial, where the divisor is of a lower degree, results in two polynomials: the quotient and the remainder.

A relationship between the zeroes and coefficients of a polynomial exists, indicating that the zeroes can be determined from the coefficients of the polynomial.

The zeroes of a quadratic polynomial are related to its coefficients. This relationship can be used to solve quadratic equations and find the zeroes of the polynomial.

An integer is called a composite number if it has more than two factors (including 1 and itself). For instance, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc. are composite numbers since each of them has more than two factors.

A positive integer greater than 1 that has exactly two factors (1 and itself) is called a prime number. For example, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc. are prime numbers.

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

Euclid's division algorithm is a method for finding the Highest Common Factor (HCF) of two positive integers. It states that any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.

The Highest Common Factor (HCF) of two or more integers is the greatest common divisor (GCD) of the given integers. For example, the HCF of 12 and 18 is 6.

The Least Common Multiple (LCM) of two or more integers is the smallest common multiple (SCM) of the given integers. For example, the LCM of 12 and 18 is 36.

A rational number is a number that can be expressed as a fraction p/q, where p and q are integers, and q is not equal to 0.

An irrational number is a number that cannot be expressed as a fraction p/q, where p and q are integers, and q is not equal to 0. It's a decimal that goes on forever without repeating.

The sum of the roots (or zeroes) of a quadratic equation ax² + bx + c = 0 is given by -b/a.

The product of the roots (or zeroes) of a quadratic equation ax² + bx + c = 0 is given by c/a.

A quadratic polynomial can be factored into two linear factors: (x - α) and (x - β), where α and β are the roots of the polynomial.

If the roots of a quadratic polynomial are α and β, the polynomial can be expressed as k(x - α)(x - β), where k is a constant.

The coefficients of a quadratic polynomial are related to the sum and product of its roots.

The roots of a quadratic polynomial are the values of x for which the polynomial evaluates to zero.

The process of finding the roots of a quadratic polynomial by splitting the middle term involves rewriting the polynomial in a way that allows it to be factored into two linear factors.

The part of a quadratic equation that determines the nature of its roots (real, equal, or no real roots). It's calculated as b² - 4ac, where a, b, and c are the coefficients of the quadratic equation.

A quadratic equation has two distinct real roots if the discriminant (b² - 4ac) is greater than zero.

A quadratic equation has two equal real roots, also known as a 'double root' if the discriminant (b² - 4ac) is equal to zero.

A quadratic equation has no real roots if the discriminant (b² - 4ac) is less than zero.

The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are coefficients and a is not equal to zero.

The quadratic formula is used to solve for the roots of a quadratic equation in standard form. It's x = (-b ± √(b² - 4ac)) / 2a.

The roots of a quadratic equation are the values of x that make the equation true. They represent the points where the graph of the equation intersects the x-axis.

The nature of roots refers to whether the roots are real, equal, or complex. It is determined by the discriminant of the quadratic equation.