Grade 10 Mathematics

In arithmetic sequences, each term is found by adding a constant difference (d) to the previous term. Example: 2, 5, 8, 11... (d = 3). In geometric sequences, each term is found by multiplying the previous term by a constant ratio (r). Example: 2, 6, 18, 54... (r = 3). The nth term formula for arithmetic is an = a1 + (n-1)d, while for geometric it is an = a1 × r^(n-1).

Synthetic division is a shortcut for dividing polynomials by (x - c). Steps: (1) Write the coefficients of the dividend, (2) Write c (the zero of divisor) on the left, (3) Bring down the first coefficient, (4) Multiply by c and add to next coefficient, (5) Repeat until done. The last number is the remainder. Example: Dividing x³ - 2x² + 4 by (x - 2): coefficients are 1, -2, 0, 4. The result gives quotient coefficients and remainder.

The Remainder Theorem states that when a polynomial P(x) is divided by (x - c), the remainder equals P(c). The Factor Theorem is a special case: (x - c) is a factor of P(x) if and only if P(c) = 0. These are essential for finding roots and factoring higher-degree polynomials. Example: If P(x) = x³ - 6x² + 11x - 6, and P(1) = 0, then (x - 1) is a factor.

To graph polynomial functions: (1) Find the degree and leading coefficient to determine end behavior, (2) Find x-intercepts by setting P(x) = 0, (3) Find y-intercept by evaluating P(0), (4) Determine multiplicity of zeros (even multiplicity = touches x-axis, odd = crosses), (5) Plot key points and sketch. For example, f(x) = x³ - 4x crosses at x = -2, 0, 2 and has end behavior: down-left, up-right.

Key circle theorems: (1) Central angle equals twice the inscribed angle on the same arc, (2) Inscribed angles on the same arc are equal, (3) Angle in a semicircle is 90°, (4) Tangent-radius is perpendicular, (5) Two tangents from external point are equal length, (6) Power of a Point: for intersecting chords (a×b = c×d), for secants from external point, and for tangent-secant relationships.

The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. To convert from general form (x² + y² + Dx + Ey + F = 0), complete the square for both x and y terms. Example: x² + y² - 6x + 4y - 12 = 0 becomes (x - 3)² + (y + 2)² = 25, center (3, -2), radius 5.

Permutation (nPr) counts arrangements where ORDER MATTERS. Formula: n!/(n-r)!. Example: Arranging 3 books from 5 = 5P3 = 60 ways. Combination (nCr) counts selections where ORDER DOES NOT MATTER. Formula: n!/[r!(n-r)!]. Example: Choosing 3 books from 5 = 5C3 = 10 ways. Remember: "Permutation = Position matters, Combination = Collection only."

These are measures of position in statistics. Quartiles divide data into 4 equal parts: Q1 (25%), Q2/median (50%), Q3 (75%). Deciles divide into 10 parts (D1 = 10%, D5 = 50%, D9 = 90%). Percentiles divide into 100 parts (P25 = Q1, P50 = median). To find the position: Pk position = k(n+1)/100 for percentiles. These help analyze data distribution and identify outliers.