SSC-Standard Mathematics for Forest Posts · Chapter 2

Algebra I: Polynomials, Remainder Theorem and Indices

What to remember

Remainder theorem: when a polynomial p(x) is divided by (x − a), the remainder is p(a). Factor theorem: (x − a) is a factor of p(x) if and only if p(a) = 0.
Identities save time: (a + b)² = a² + 2ab + b², a² − b² = (a + b)(a − b), a³ + b³ = (a + b)(a² − ab + b²), and a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

1. Polynomials: basic terms

A polynomial in x is a sum of terms of the form c·xⁿ, where n is a whole number. Example: 3x³ − 5x² + 2x − 7.

Degree: highest power of x (here 3). Degree 1 is linear, 2 quadratic, 3 cubic.
Coefficient: the number multiplying a power of x. Constant term: the term without x.
Zero (root) of p(x): a value a such that p(a) = 0.
A non-zero constant has degree 0. The degree of the zero polynomial is not defined.
A polynomial of degree n has at most n real zeros.

Type	Example
Monomial	5x²
Binomial	x + 3
Trinomial	x² − 4x + 4
Expression such as x + 1/x	Not a polynomial (negative power)
Expression such as √x + 2	Not a polynomial (fractional power)

Value of a polynomial. To find p(2) for p(x) = x³ − 3x + 1, put x = 2: 8 − 6 + 1 = 3.

Zeros of a quadratic. For ax² + bx + c = 0: sum of roots = −b/a; product of roots = c/a. Roots by formula: x = [−b ± √(b² − 4ac)] / 2a. The expression b² − 4ac is the discriminant: positive means two distinct real roots, zero means equal roots, negative means no real roots.

Example: x² − 5x + 6 = 0 has roots 2 and 3: sum 5, product 6.

2. Remainder theorem and factor theorem

Remainder theorem. If p(x) is divided by (x − a), the remainder is p(a). If divided by (ax − b), the remainder is p(b/a).

Worked examples:

1. Remainder of x³ − 2x² + x − 5 divided by (x − 2): p(2) = 8 − 8 + 2 − 5 = −3.
2. Remainder of x² + 3x + 2 divided by (x + 1): p(−1) = 1 − 3 + 2 = 0, so (x + 1) is a factor.
3. Remainder of 2x³ + x² − 5 divided by (x + 2): p(−2) = −16 + 4 − 5 = −17.
4. Remainder of x⁴ + 1 divided by (x − 1): p(1) = 2.

Factor theorem. (x − a) is a factor of p(x) exactly when p(a) = 0.

Worked examples:

5. If (x − 2) is a factor of x² + kx − 10, then 4 + 2k − 10 = 0, so k = 3.
6. Find k if x³ − kx² + 2x + 6 is divisible by (x + 1): p(−1) = −1 − k − 2 + 6 = 0, so k = 3.
7. Factorise x³ − 6x² + 11x − 6. Test x = 1: 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor. Dividing gives x² − 5x + 6 = (x − 2)(x − 3). So the factors are (x − 1)(x − 2)(x − 3).

Tip for finding rational roots. If the leading coefficient is 1, try factors of the constant term (± values). Add the coefficients: if the sum is 0, then x = 1 is a root; if the coefficients of the even powers add up to the same total as the coefficients of the odd powers, then x = −1 is a root.

Remainder with two divisors. If p(x) leaves remainder 5 when divided by (x − 1) and 7 when divided by (x − 2), write p(1) = 5 and p(2) = 7, then form equations for the unknown coefficients.

3. Algebraic identities

Identity	Result
(a + b)²	a² + 2ab + b²
(a − b)²	a² − 2ab + b²
a² − b²	(a + b)(a − b)
(a + b)³	a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b)
(a − b)³	a³ − 3a²b + 3ab² − b³ = a³ − b³ − 3ab(a − b)
a³ + b³	(a + b)(a² − ab + b²)
a³ − b³	(a − b)(a² + ab + b²)
(a + b + c)²	a² + b² + c² + 2(ab + bc + ca)
a³ + b³ + c³ − 3abc	(a + b + c)(a² + b² + c² − ab − bc − ca)
(x + a)(x + b)	x² + (a + b)x + ab

Special results:

If a + b + c = 0, then a³ + b³ + c³ = 3abc.
If x + 1/x = k, then x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k.
If x − 1/x = k, then x² + 1/x² = k² + 2.
(a + b)² − (a − b)² = 4ab; (a + b)² + (a − b)² = 2(a² + b²).

Worked examples:

1. If x + 1/x = 5, then x² + 1/x² = 25 − 2 = 23 and x³ + 1/x³ = 125 − 15 = 110.
2. If a + b = 7 and ab = 12, then a² + b² = 49 − 24 = 25 and a³ + b³ = 343 − 3 × 12 × 7 = 343 − 252 = 91.
3. 103² = (100 + 3)² = 10,000 + 600 + 9 = 10,609.
4. 97 × 103 = (100 − 3)(100 + 3) = 10,000 − 9 = 9,991.
5. If a = 4, b = 3, c = −7, then a + b + c = 0, so a³ + b³ + c³ = 3abc = 3 × 4 × 3 × (−7) = −252.
6. Factorise x² − 9x + 20 = (x − 4)(x − 5). Factorise 2x² + 7x + 3 = (2x + 1)(x + 3).

4. Indices (exponents)

Law	Statement
Product	aᵐ × aⁿ = aᵐ⁺ⁿ
Quotient	aᵐ / aⁿ = aᵐ⁻ⁿ
Power of a power	(aᵐ)ⁿ = aᵐⁿ
Product to a power	(ab)ⁿ = aⁿbⁿ
Quotient to a power	(a/b)ⁿ = aⁿ/bⁿ
Zero power	a⁰ = 1 (a ≠ 0)
Negative power	a⁻ⁿ = 1/aⁿ
Fractional power	a^(1/n) = nth root of a; a^(m/n) = (nth root of a)ᵐ

Note: (aᵐ)ⁿ means a to the power (m × n), but a^(mⁿ) is different. Compare (2²)³ = 2⁶ = 64 and 2^(2³) = 2⁸ = 256.

Worked examples:

1. 2⁵ × 2³ = 2⁸ = 256.
2. 3⁷ / 3⁴ = 3³ = 27.
3. (5²)³ = 5⁶ = 15,625.
4. 4⁻² = 1/16.
5. 8^(2/3) = (∛8)² = 2² = 4. 27^(−1/3) = 1/3. 16^(3/4) = 8.
6. If 2^(x+1) = 32, then x + 1 = 5 and x = 4.
7. If 3^(2x) = 81, then 2x = 4 and x = 2.
8. Simplify (2ˣ⁺² − 2ˣ)/2ˣ = 2² − 1 = 3.
9. If aˣ = aʸ and a ≠ 0, 1, −1, then x = y.
10. (x^(a−b))^(a+b) = x^(a²−b²), because the powers multiply.

Surds. √a × √b = √(ab); √a / √b = √(a/b). Rationalise 1/(√a + √b) by multiplying by (√a − √b)/(√a − √b): result (√a − √b)/(a − b). Example: 1/(√3 + √2) = √3 − √2. A useful result: (√3 + √2)(√3 − √2) = 1.

Standard values: 2¹⁰ = 1,024; 3⁴ = 81; 5³ = 125; 7² = 49; 11² = 121; 2⁷ = 128; 2⁸ = 256.

5. Simple equations using algebra

Linear equation: 3x + 5 = 20 gives x = 5.
Quadratic by factorisation: x² − 7x + 12 = 0 gives (x − 3)(x − 4) = 0, so x = 3 or 4.
Pair of equations: x + y = 10, x − y = 2 gives x = 6, y = 4.
Use substitution to check each answer.

Exam traps

The remainder for (x + a) is p(−a), not p(a).
For (ax − b), put x = b/a.
A polynomial cannot contain 1/x or √x terms.
The sum of roots of ax² + bx + c is −b/a (negative sign).
a³ + b³ + c³ = 3abc only when a + b + c = 0.
a⁰ = 1 for every non-zero a.
(a + b)² is not a² + b².
2^(2³) (= 256) is different from (2²)³ (= 64).

One-liners

1. Remainder theorem: remainder of p(x) ÷ (x − a) is p(a).
2. Factor theorem: p(a) = 0 means (x − a) is a factor.
3. a² − b² = (a + b)(a − b).
4. a³ + b³ = (a + b)(a² − ab + b²).
5. x + 1/x = k gives x² + 1/x² = k² − 2.
6. Sum of roots = −b/a; product = c/a.
7. Discriminant = b² − 4ac.
8. aᵐ × aⁿ = aᵐ⁺ⁿ.
9. a⁻ⁿ = 1/aⁿ.
10. 8^(2/3) = 4.
11. If a + b + c = 0, a³ + b³ + c³ = 3abc.
12. 1/(√3 + √2) = √3 − √2.

Practice questions

The degree of the polynomial 3x^3 - 5x^2 + 2x - 7 is:
1. 2
2. 4
3. 3
4. 7
Answer
C. 3
The highest power of x is 3.
Which of the following is a polynomial?
1. square root of x + 2
2. x^2 + 3x + 1
3. x + 1/x
4. x^-2 + 1
Answer
B. x^2 + 3x + 1
Polynomials have only whole-number powers of x.
The remainder when x^3 - 2x^2 + x - 5 is divided by (x - 2) is:
1. 3
2. -5
3. 5
4. -3
Answer
D. -3
p(2) = 8 - 8 + 2 - 5 = -3.
The remainder when 2x^3 + x^2 - 5 is divided by (x + 2) is:
1. -7
2. -17
3. -5
4. 17
Answer
B. -17
p(-2) = -16 + 4 - 5 = -17.
The remainder when x^4 + 1 is divided by (x - 1) is:
1. 0
2. 2
3. 1
4. 4
Answer
B. 2
p(1) = 1 + 1 = 2.
The remainder when x^3 - 3x^2 + 4x - 7 is divided by (x - 3) is:
1. 12
2. 0
3. -7
4. 5
Answer
D. 5
p(3) = 27 - 27 + 12 - 7 = 5.
The remainder when 2x^2 - 3x + 1 is divided by (2x - 1) is:
1. 0
2. 1
3. -1
4. 1/2
Answer
A. 0
Put x = 1/2: 1/2 - 3/2 + 1 = 0.
The remainder when x^51 + 51 is divided by (x + 1) is:
1. 52
2. 51
3. 50
4. 0
Answer
C. 50
p(-1) = -1 + 51 = 50.
If (x - 2) is a factor of x^2 + kx - 10, then k is:
1. -3
2. 5
3. 3
4. -5
Answer
C. 3
4 + 2k - 10 = 0 gives k = 3.
If x^3 - kx^2 + 2x + 6 is divisible by (x + 1), then k is:
1. 3
2. -3
3. 2
4. 6
Answer
A. 3
p(-1) = -1 - k - 2 + 6 = 0 gives k = 3.
Which of the following is a factor of x^3 - 6x^2 + 11x - 6?
1. x - 6
2. x + 3
3. x + 1
4. x - 3
Answer
D. x - 3
p(3) = 27 - 54 + 33 - 6 = 0.
The value of x^3 - 3x + 1 at x = 2 is:
1. 1
2. 3
3. 5
4. 7
Answer
B. 3
8 - 6 + 1 = 3.
The sum of the roots of 2x^2 - 8x + 6 = 0 is:
1. -4
2. 4
3. 3
4. 8
Answer
B. 4
Sum = -b/a = 8/2 = 4.
The product of the roots of 3x^2 - 7x + 12 = 0 is:
1. 7/3
2. -4
3. -7/3
4. 4
Answer
D. 4
Product = c/a = 12/3 = 4.
The quadratic x^2 - 6x + 9 = 0 has:
1. Two equal real roots
2. Two distinct real roots
3. No real roots
4. Two imaginary roots only
Answer
A. Two equal real roots
Discriminant = 36 - 36 = 0.
If a + b = 7 and ab = 12, then a^2 + b^2 is:
1. 37
2. 19
3. 25
4. 49
Answer
C. 25
49 - 2 x 12 = 25.
If a + b = 7 and ab = 12, then a^3 + b^3 is:
1. 91
2. 127
3. 169
4. 343
Answer
A. 91
a^3 + b^3 = 343 - 3 x 12 x 7 = 91.
If a + b = 10 and a - b = 2, then ab is:
1. 20
2. 48
3. 12
4. 24
Answer
D. 24
4ab = 100 - 4 = 96, so ab = 24.
If x + 1/x = 5, then x^2 + 1/x^2 is:
1. 27
2. 25
3. 23
4. 21
Answer
C. 23
25 - 2 = 23.
If x + 1/x = 3, then x^3 + 1/x^3 is:
1. 18
2. 27
3. 9
4. 24
Answer
A. 18
k^3 - 3k = 27 - 9 = 18.
If x - 1/x = 4, then x^2 + 1/x^2 is:
1. 16
2. 18
3. 14
4. 20
Answer
B. 18
16 + 2 = 18.
The value of 103 squared is:
1. 10,069
2. 10,906
3. 10,309
4. 10,609
Answer
D. 10,609
(100 + 3)^2 = 10,000 + 600 + 9.
The value of 97 x 103 is:
1. 10,009
2. 9,999
3. 9,991
4. 9,901
Answer
C. 9,991
(100 - 3)(100 + 3) = 10,000 - 9.
The value of 99 squared minus 98 squared is:
1. 1
2. 99
3. 197
4. 196
Answer
C. 197
(99 + 98)(99 - 98) = 197.
If a = 4, b = 3 and c = -7, then a^3 + b^3 + c^3 equals:
1. -252
2. 252
3. 0
4. -84
Answer
A. -252
a + b + c = 0, so the sum is 3abc = 3 x 4 x 3 x (-7) = -252.
The value of 2^5 x 2^3 is:
1. 512
2. 256
3. 128
4. 64
Answer
B. 256
2^8 = 256.
The value of 3^7 / 3^4 is:
1. 9
2. 81
3. 3
4. 27
Answer
D. 27
3^3 = 27.
The value of (5^2)^3 is:
1. 3,125
2. 390,625
3. 625
4. 15,625
Answer
D. 15,625
5^6 = 15,625.
The value of 4^-2 is:
1. 1/16
2. 1/8
3. -8
4. -16
Answer
A. 1/16
4^-2 = 1/4^2 = 1/16.
The value of 8^(2/3) is:
1. 16
2. 2
3. 8/3
4. 4
Answer
D. 4
Cube root of 8 is 2; 2^2 = 4.
The value of 27^(-1/3) is:
1. 1/3
2. 3
3. -3
4. 1/9
Answer
A. 1/3
27^(1/3) = 3, so the reciprocal is 1/3.
The value of 16^(3/4) is:
1. 12
2. 4
3. 8
4. 64
Answer
C. 8
Fourth root of 16 is 2; 2^3 = 8.
If 2^(x+1) = 32, then x is:
1. 5
2. 3
3. 4
4. 16
Answer
C. 4
x + 1 = 5.
If 3^(2x) = 81, then x is:
1. 2
2. 4
3. 3
4. 9
Answer
A. 2
2x = 4.
The value of (2^(x+2) - 2^x) / 2^x is:
1. 2
2. 4
3. 1
4. 3
Answer
D. 3
2^2 - 1 = 3.
If 5^(x-1) = 1/125, then x is:
1. -3
2. -2
3. 4
4. 2
Answer
B. -2
x - 1 = -3, so x = -2.
The value of 2^(2^3) is:
1. 64
2. 256
3. 16
4. 32
Answer
B. 256
2^3 = 8 and 2^8 = 256.
The value of 1/(sqrt3 + sqrt2) after rationalising is:
1. sqrt3 + sqrt2
2. 1/(sqrt3 - sqrt2)
3. sqrt3 - sqrt2
4. sqrt3 x sqrt2
Answer
C. sqrt3 - sqrt2
Multiply by (sqrt3 - sqrt2); the denominator becomes 3 - 2 = 1.
The value of (sqrt3 + sqrt2)(sqrt3 - sqrt2) is:
1. 1
2. 5
3. sqrt6
4. 2 sqrt3
Answer
A. 1
3 - 2 = 1.
The value of sqrt12 x sqrt3 is:
1. 36
2. 6
3. sqrt15
4. 3 sqrt3
Answer
B. 6
sqrt36 = 6.
The roots of x^2 - 7x + 12 = 0 are:
1. 2 and 6
2. -3 and -4
3. 3 and 4
4. 1 and 12
Answer
C. 3 and 4
(x - 3)(x - 4) = 0.
Consider the statements: 1. The remainder when p(x) is divided by (x + a) is p(-a). 2. The remainder when p(x) is divided by (x + a) is p(a). Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Put x = -a for the divisor (x + a).
Consider the statements: 1. a^3 + b^3 + c^3 = 3abc for all values of a, b and c. 2. If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
The result holds only when a + b + c = 0.
Consider the statements: 1. a^0 = 1 for every non-zero a. 2. a^-n = 1/a^n. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are standard laws of indices.
Consider the statements: 1. (a + b)^2 = a^2 + b^2. 2. 2^(2^3) equals (2^2)^3. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
(a + b)^2 has a 2ab term; 2^(2^3) = 256 while (2^2)^3 = 64.

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1.The difference between a number and the square root of the number is 2. The number is:
2.The difference between two numbers is 8. If twice the larger number is 4 more than three times the smaller number, what is the larger number?
3.The expression x4 – 8x2 + m will be a perfect square when the value of m is:
4.The expression √(10 + 2(√6 - √15 - √10)) is equal to:
5.Seven years ago Ramu was five times as old as his son. The present age difference between Ramu and his son is 28 years. Find the sum of their present ages.
6.The ages of two siblings add up to 44 years. If one sibling is 4 years younger than twice the other's age, what is the difference between their ages?
7.The ages of a mother and daughter are 36 years and 12 years, respectively. In how many years will the mother be twice the age of the daughter?
8.Simplify the expression (36p² + 49q²) (6p + 7q) (6p – 7q)
9.Simplify (2pq + 11q)2 - (2pq - 11q)2.
10.The sum of two positive numbers is 26 and their product is 144. The positive difference between them is:

10 left. You can check any time.