The document discusses binomial theorem, quadratic equations, and progressions. 1) The binomial theorem describes the expansion of terms like (a + b)n and gives the formula for finding individual terms. It is illustrated using the expansion of (a - b)7. 2) Quadratic equations of the form ax2 + bx + c = 0 can be solved using the quadratic formula. The sum and product of the roots are also given. 3) Progressions include arithmetic, geometric, and harmonic sequences. Arithmetic progressions have a constant difference between terms while geometric progressions multiply the previous term by a fixed ratio. Formulas are given for the nth term and sum of finite and infinite progress

1. Topic: BINOMIAL THEOREM AND PROGRESSIONS
BINOMIAL THEOREM
Binomial theorem describes the expansions of terms as (a + b)n into terms involving axby, where the sum of exponent x and y is n.
Consider: (a + b)n
n(n − 1)an−2b 2 n(n − 1)(n − 2)an−3b 3
(a + b)n = an + nan−1b + + + ... + b n
2! 3!
Pascal’s Triangle
1
121
1331
14641
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
Illustration:
Expand: (a – b)7
a7 − 7a6b + 21a5b2 − 35a4b3 + 35a3b 4 − 21a2b5 + 7ab6 − b2
Observations:
1. Number of expansion is n + 1
2. First term is an
3. Last term is bn
4. The exponent of a is decreasing by 1 from left to right
5. The exponent of b is increasing from left to right.
6. Sum of exponents of a and b is equal to n.
7. Exponent of b is 1 lesser than the position number
8. Coefficient of terms in the expansion is symmetrical.
Finding a particular term in a Binomial expansion
n(n − 1)(n − r + 2)an−r +1b r −1
r th term =
(r − 1)!
r th term = nCr −1an−r +1br −1
Illustration:
Find the 5th term of expansion (a – b)7
5th term = 7C5 −1a7 − 5 +1b5 −1
5th term = 35a3b 4
Sum of coefficients
Illustration:
Find the sum of coefficients of expansion (x – 2y)7
Substitute 1 to x and y: (1 – 2)7
Sum of coefficient is -1
QUADRATIC EQUATION OF THE SECOND ORDER
ax 2 + bx + c = 0
− b ± b 2 − 4ac
The roots, x1 and x2 can be found using Quadratic Formula: x =
2a
Sum of Roots
−b
x1 + x 2 =
a
Products of Roots
c
x1 * x 2 =
a
Discriminants
Discriminants describes the real nature of a polynomial’s root.
D = b2 − 4ac
Recall:
Rational numbers: numbers that can be expressed as quotient of two integers. 2/3, 2 -5, 0.1212…, 0.4444…
Irrational numbers: real numbers that cannot be expressed as fractions. 2 , π, e
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2. Topic: BINOMIAL THEOREM AND PROGRESSIONS
D = b2 − 4ac Nature of roots
D=0 Equal
D>0 not perfect square Unequal, Real, Irrational
D>0 perfect square Unequal, Real, Rational
D<0 Unequal, Imaginary
Remainder Theorem
If a polynomial f(x) is divided by (x – r), the remainder is equal to f(r).
f(x) R
= Q(x) + , Q=quotient and R=remainder
x −r x −r
f(x) = Q(x)(x − r) + R ; let x=r
f(r) = Qr(r − r) + R
f(r) = R
Illustration:
Find the remainder when x 4 − 5x 3 + 7x 2 − 3x + 6 is divided by x + 1.
R = f(r); r = -1
R = f(-1) = (-1)4 – 5(-1)3 + 7(-1)2 – 3(-1) + 6
R = 22
PROGRESSIONS
Arithmetic Progressions
Arithmetic Progressions also known as arithmetic sequence is a series of numbers such that the difference of any successive
members of the sequence is a constant.
1. nth term of Arithmetic Progressions
an = a1 + (n − 1)d
2. Sum of Arithmetic Progressions
S=
n
2
(a1 + an ) S=
n
2
[ 2a
1 ]
+ (n − 1) d
3. Arithmetic Mean
Arithmetic mean or simply “average”, is the sum of two or more terms and then dividing by the number of terms.
a1 + a2
Am =
2
Geometric Progression
Geometric Progressions also known as geometric sequence is a series of numbers where the term after the first is found by
multiplying the previous one by a fixed non-zero number called the common ratio. For example: 2, 6, 18, 54…the common ratio is 3.
1. nth term of Geometric Progressions
a2 a
an = a1r n−1 r = common ratio = = 3
a1 a2
2. Sum of Geometric Progressions
If r < 1.0
a1 + ran a1 (1 − r n )
S= S=
1−r 1−r
If r > 1.0
ra + a1 a1 (r n − 1)
S= n S=
r −1 r −1
3. Geometric Mean
Gm = n a1a2...an
Infinite Geometric Progressions
Infinite geometric series is an infinite series whose successive terms have a common ratio.
a1
S=
1−r
Harmonic Progressions
Harmonic progressions is a progressions formed by taking the reciprocals of Arithmetic Progression.
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