CBSE Class 8 Mathematics Algebraic Expressions And Identities Worksheet Set 04

Question. If a = 7 and b = 3. Find a² – b².
Answer : 40

Question. What should be subtracted from (x + y)² to get (x² + y²).
Answer : 2xy

Question. If a = 5 and b = 3. Find the value of (x + y)² – (x² + y²).
Answer : 30

Question. Multiply and give the answer in simplified form (z – 6) (z – 5).
Answer : z² – 11z + 30

Question. Find the product of 9xy² and 4x³y.
Answer : 36x⁴y³

Question. Add : 7xy + 5yz – 3zx, – 5yz + 3zx, –6xy + yz + zx.
Answer : xy + yz + zx

Question. What should be added to a² + b² to get (a – b)².
Answer : –2ab

Question. Find the volume of a rectangular box with length, breadth and height respectively 2ax, 3cy and 5gz.
Answer : 30 acgxyz

Question. Find the area of a rectangle with length and breadth respectively 5mn and 3ns.
Answer : 15mn²s

Question. Simplify : (5x²p)³
Answer : 125x⁶p³

Question. Find the value of (5x + 7y) if x = 3 and y = 2.
Answer : 29

Question. How much is (√19 + √17) (√19 – √17) ?
Answer : 2

Question. Find (4/3 m + 3/4 n) (4/3 m - 3/4 n) 
Answer : 16/9 m² - 9/16 n²

Question. Find the value of (2/3 m - 3/2 n)² + 2mn
Answer : 4/9 m² + 9/4 n²

Question. Subtract : 9a – 6ab + 3b + 5 from 12a – 9ab + 5b – 3.
Answer : 3a – 3ab + 2b – 8

Question. If x = 3 and y = 5. Find x² + y².
Answer : 34

Question. Add ab – bc, bc – ca, ca – ab.
Answer : 0

Question. Identify the coefficients in the algebric expression 3xyz² – 2zy.
Answer : (3, –2)

Question. Find the value of 47² – 43².
Answer : 360

Question. Find the product of (3n – 5p) (3n + 5p)
Answer : 9n² – 25p²

1. Classify the following expressions as monomials, binomials and trinomials.
\(4a^2 \quad \text{________} \qquad 5a^2b + 3 \quad \text{________}\)
\(7b^3 \quad \text{________} \qquad 8a^2 + 7ab + b^2 \quad \text{________}\)
\(4x^2 + 1 \quad \text{________} \qquad a^2 + b^2 \quad \text{________}\)

Answer:
\(4a^2\): Monomial
\(5a^2b + 3\): Binomial
\(7b^3\): Monomial
\(8a^2 + 7ab + b^2\): Trinomial
\(4x^2 + 1\): Binomial
\(a^2 + b^2\): Binomial
In simple words: Expressions with 1 term are monomials, 2 terms are binomials, and 3 terms are trinomials.

Exam Tip: Count the number of unlike terms separated by plus or minus signs to determine the type of polynomial.

2. Write the degrees of the following polynomials.
a) \(2x^2 + 3xy + 5y^2 + 2 \quad \text{________}\)
b) \(7x^2y + 3xy^2 + 4xy \quad \text{________}\)
c) \(x^3y^3 - 4x^2y + 5xy^2 + x^2y^2 \quad \text{________}\)
d) \(4x^5 + 5x^3 + 7x^2 + 2 \quad \text{________}\)

Answer:
a) \(2x^2 + 3xy + 5y^2 + 2\) - Degree is 2
b) \(7x^2y + 3xy^2 + 4xy\) - Degree is 3
c) \(x^3y^3 - 4x^2y + 5xy^2 + x^2y^2\) - Degree is 6
d) \(4x^5 + 5x^3 + 7x^2 + 2\) - Degree is 5
In simple words: The degree of a polynomial is the highest sum of the powers of variables in any single term.

Exam Tip: In terms with multiple variables, add the exponents of all variables in that term to find its degree.

3. Add.
a) \((10x^2 + 5x - 3) + (7x^2 - 2x + 7) =\)
b) \((3x^2y + 4x^2y^2 - 7xy^2) + (9x^2y - x^2y^2 + 3xy^2) =\)
c) \((10x^2y - 3xy^2 + 5x^2y^2 + 22) + (3x^2y^2 + 7) =\)

Answer:
a) Grouping like terms:
\(= (10x^2 + 7x^2) + (5x - 2x) + (-3 + 7)\)
\(= 17x^2 + 3x + 4\)

b) Grouping like terms:
\(= (3x^2y + 9x^2y) + (4x^2y^2 - x^2y^2) + (-7xy^2 + 3xy^2)\)
\(= 12x^2y + 3x^2y^2 - 4xy^2\)

c) Grouping like terms:
\(= 10x^2y - 3xy^2 + (5x^2y^2 + 3x^2y^2) + (22 + 7)\)
\(= 10x^2y - 3xy^2 + 8x^2y^2 + 29\)
In simple words: Combine the coefficients of terms having the exact same variables and exponents.

Exam Tip: Arrange like terms together carefully by checking signs to avoid calculation errors.

4. Subtract \(2x^2 + 5x - 7\) from \(7x^2 - 5x + 3\).

Answer:
\(= (7x^2 - 5x + 3) - (2x^2 + 5x - 7)\)
\(= 7x^2 - 5x + 3 - 2x^2 - 5x + 7\)
\(= (7x^2 - 2x^2) + (-5x - 5x) + (3 + 7)\)
\(= 5x^2 - 10x + 10\)
In simple words: Distribute the negative sign to all terms inside the second bracket before combining like terms.

Exam Tip: 'Subtract A from B' always means \(B - A\).

5. Subtract \(2ab + 7a^2 + 8b^2\) from \(10b^2 + 7ab + a^2\).

Answer:
\(= (10b^2 + 7ab + a^2) - (2ab + 7a^2 + 8b^2)\)
\(= 10b^2 + 7ab + a^2 - 2ab - 7a^2 - 8b^2\)
\(= (a^2 - 7a^2) + (7ab - 2ab) + (10b^2 - 8b^2)\)
\(= -6a^2 + 5ab + 2b^2\)
In simple words: Write the second expression first, then subtract each term of the first expression by changing its sign.

Exam Tip: Double-check the signs of each term when removing brackets.

1. Write the degrees of the following polynomials.
a) \(4xy + 2x^2y + 3y^2 + 5x^2y^2 \quad \text{________}\) \qquad b) \(4xy + 3 \quad \text{________}\)
c) \(7x + 2 \quad \text{________}\) \qquad d) \(32 + 22xy + 3x^2 + 7y^2 + x^3y^2 \quad \text{________}\)

Answer:
a) \(4xy + 2x^2y + 3y^2 + 5x^2y^2\) - Degree is 4
b) \(4xy + 3\) - Degree is 2
c) \(7x + 2\) - Degree is 1
d) \(32 + 22xy + 3x^2 + 7y^2 + x^3y^2\) - Degree is 5
In simple words: Find the highest sum of powers of variables in any single term.

Exam Tip: A constant term has a degree of 0.

2. Tick the pairs of like terms.
a) \(3a^2b, -5ba^2 \quad [\;\;]\) \qquad b) \(7abc, 8ab \quad [\;\;]\)
c) \(15xy, -7yz \quad [\;\;]\) \qquad d) \(5a^3b^2c, 12b^2ca^3 \quad [\;\;]\)

Answer:
a) \(3a^2b, -5ba^2\) - [✓] (Like terms)
b) \(7abc, 8ab\) - [   ] (Unlike terms)
c) \(15xy, -7yz\) - [   ] (Unlike terms)
d) \(5a^3b^2c, 12b^2ca^3\) - [✓] (Like terms)
In simple words: Like terms must have the exact same variables with the exact same exponents, regardless of order.

Exam Tip: Multiplication is commutative, so \(a^2b\) is identical to \(ba^2\).

3. Add the following.
a) \(3a^2 + 4ab - b^2, \; 7b^2 - 4ab + 2a^2, \; a^2 + b^2\)
b) \(5xy - 7x^2 - 3y^2 + 4x^2y, \; 5x^2y - xy + x^2 + y^2\)
c) \(5a^3 - 2b^3 + 3a^2b + 7ab^2, \; 3a^2b - 5ab^2, \; a^3 + b^3 - a^2b\)

Answer:
a) \((3a^2 + 4ab - b^2) + (7b^2 - 4ab + 2a^2) + (a^2 + b^2)\)
\(= (3a^2 + 2a^2 + a^2) + (4ab - 4ab) + (-b^2 + 7b^2 + b^2)\)
\(= 6a^2 + 7b^2\)

b) \((5xy - 7x^2 - 3y^2 + 4x^2y) + (5x^2y - xy + x^2 + y^2)\)
\(= (-7x^2 + x^2) + (-3y^2 + y^2) + (5xy - xy) + (4x^2y + 5x^2y)\)
\(= -6x^2 - 2y^2 + 4xy + 9x^2y\)

c) \((5a^3 - 2b^3 + 3a^2b + 7ab^2) + (3a^2b - 5ab^2) + (a^3 + b^3 - a^2b)\)
\(= (5a^3 + a^3) + (-2b^3 + b^3) + (3a^2b + 3a^2b - a^2b) + (7ab^2 - 5ab^2)\)
\(= 6a^3 - b^3 + 5a^2b + 2ab^2\)
In simple words: Combine all terms with identical variable parts.

Exam Tip: Cross off terms as you add them to ensure none are missed.

4. Subtract \(3x^2 - 4y^2\) from the sum of \(x^2 + y^2 - 2xy\) and \(3x^2 - 4xy + 7y^2\).

Answer:
Step 1: Find the sum:
\(\text{Sum} = (x^2 + y^2 - 2xy) + (3x^2 - 4xy + 7y^2)\)
\(= (x^2 + 3x^2) + (y^2 + 7y^2) + (-2xy - 4xy)\)
\(= 4x^2 + 8y^2 - 6xy\)
Step 2: Subtract \(3x^2 - 4y^2\) from the sum:
\(= (4x^2 + 8y^2 - 6xy) - (3x^2 - 4y^2)\)
\(= 4x^2 + 8y^2 - 6xy - 3x^2 + 4y^2\)
\(= (4x^2 - 3x^2) + (8y^2 + 4y^2) - 6xy\)
\(= x^2 + 12y^2 - 6xy\)
In simple words: First add the two expressions, then subtract the given expression from that sum.

Exam Tip: Perform addition first, simplify completely, then perform the subtraction.

5. Subtract \(4a^3 - 3a^2b + ab^2 - b^3\) from \(a^3 + b^3 - 3a^2b + 7ab^2\).

Answer:
\(= (a^3 + b^3 - 3a^2b + 7ab^2) - (4a^3 - 3a^2b + ab^2 - b^3)\)
\(= a^3 + b^3 - 3a^2b + 7ab^2 - 4a^3 + 3a^2b - ab^2 + b^3\)
\(= (a^3 - 4a^3) + (b^3 + b^3) + (-3a^2b + 3a^2b) + (7ab^2 - ab^2)\)
\(= -3a^3 + 2b^3 + 6ab^2\)
In simple words: Subtract each term of the first polynomial from the corresponding term of the second polynomial by reversing signs.

Exam Tip: Watch out for terms that cancel out completely, like \(-3a^2b + 3a^2b\).

1. Add \(x^3 - 5x^2 + 7x + 2, \; 15x^2 + 10x - 7\) and \(x^3 - 13x + 2\).

Answer:
\(= (x^3 - 5x^2 + 7x + 2) + (15x^2 + 10x - 7) + (x^3 - 13x + 2)\)
\(= (x^3 + x^3) + (-5x^2 + 15x^2) + (7x + 10x - 13x) + (2 - 7 + 2)\)
\(= 2x^3 + 10x^2 + 4x - 3\)
In simple words: Add all like terms together across the three expressions.

Exam Tip: Group by descending powers of \(x\) to keep the final polynomial in standard form.

2. Add \(3x^2y + 4x^3y - xy^2 + x^2y^2, \; 11x^2y - x^3y + 5x^2y^2\) and \(5x^3y - x^2y^2 + 7x^2y - 3xy^2\).

Answer:
\(= (3x^2y + 4x^3y - xy^2 + x^2y^2) + (11x^2y - x^3y + 5x^2y^2) + (5x^3y - x^2y^2 + 7x^2y - 3xy^2)\)
\(= (4x^3y - x^3y + 5x^3y) + (3x^2y + 11x^2y + 7x^2y) + (-xy^2 - 3xy^2) + (x^2y^2 + 5x^2y^2 - x^2y^2)\)
\(= 8x^3y + 21x^2y - 4xy^2 + 5x^2y^2\)
In simple words: Combine coefficients of identical multi-variable terms.

Exam Tip: Be careful with signs of negative coefficients like \(-xy^2\) and \(-3xy^2\).

3. Subtract \(5a^3 - 2a^2 + 7\) from the sum of \(a^3 - 3a^2 + 5a + 1\) and \(7a^2 + a + 3\).

Answer:
Step 1: Find the sum:
\(\text{Sum} = (a^3 - 3a^2 + 5a + 1) + (7a^2 + a + 3)\)
\(= a^3 + (-3a^2 + 7a^2) + (5a + a) + (1 + 3)\)
\(= a^3 + 4a^2 + 6a + 4\)
Step 2: Subtract \(5a^3 - 2a^2 + 7\) from the sum:
\(= (a^3 + 4a^2 + 6a + 4) - (5a^3 - 2a^2 + 7)\)
\(= a^3 + 4a^2 + 6a + 4 - 5a^3 + 2a^2 - 7\)
\(= (a^3 - 5a^3) + (4a^2 + 2a^2) + 6a + (4 - 7)\)
\(= -4a^3 + 6a^2 + 6a - 3\)
In simple words: Add the first two polynomials first, then subtract the third polynomial from that total.

Exam Tip: Write all terms in standard descending order of powers before subtracting.

4. What should be added to \(x^4 + 2x^2 - 7\) to obtain \(4x^4 - 3x^3 + x^2 + 2\)?

Answer:
Required expression = \((4x^4 - 3x^3 + x^2 + 2) - (x^4 + 2x^2 - 7)\)
\(= 4x^4 - 3x^3 + x^2 + 2 - x^4 - 2x^2 + 7\)
\(= (4x^4 - x^4) - 3x^3 + (x^2 - 2x^2) + (2 + 7)\)
\(= 3x^4 - 3x^3 - x^2 + 9\)
In simple words: Subtract the given expression from the target expression.

Exam Tip: 'What should be added to A to get B' is always calculated as \(B - A\).

5. What must be subtracted from \(4x^2y^2 + 3xy + 3xy^2\) to obtain \(8xy^2 - 4x^2y + 7x^2y^2 + 10xy\)?

Answer:
Required expression = \((4x^2y^2 + 3xy + 3xy^2) - (8xy^2 - 4x^2y + 7x^2y^2 + 10xy)\)
\(= 4x^2y^2 + 3xy + 3xy^2 - 8xy^2 + 4x^2y - 7x^2y^2 - 10xy\)
\(= (4x^2y^2 - 7x^2y^2) + (3xy - 10xy) + (3xy^2 - 8xy^2) + 4x^2y\)
\(= -3x^2y^2 - 7xy - 5xy^2 + 4x^2y\)
In simple words: Subtract the target expression from the original expression.

Exam Tip: 'What must be subtracted from A to get B' is calculated as \(A - B\).

1. Find the products of the following expressions.
a) \((3x^2y) \times \left(\frac{-1}{5}xyz\right) \times \left(\frac{3}{5}y^2z\right) =\)
b) \(\left(\frac{1}{8}ab\right) \times \left(\frac{-8}{11}bc\right) \times \left(\frac{-22}{3}ca\right) =\)
c) \(\left(\frac{3}{8}x^3y\right) \times \left(\frac{-4}{7}y^2z\right) \times \left(\frac{-7}{11}z^2x\right) =\)
d) \(\left(\frac{5}{7}ab\right) \times \left(\frac{5}{7}a^2b\right) \times \left(\frac{5}{22}b^2c\right) =\)

Answer:
a) \(= 3 \times \left(\frac{-1}{5}\right) \times \left(\frac{3}{5}\right) \times x^{2+1} \times y^{1+1+2} \times z^{1+1}\)
\(= \left(\frac{-9}{25}\right) x^3 y^4 z^2\)

b) \(= \left(\frac{1}{8}\right) \times \left(\frac{-8}{11}\right) \times \left(\frac{-22}{3}\right) \times a^{1+1} \times b^{1+1} \times c^{1+1}\)
\(= \left(\frac{1 \times (-8) \times (-22)}{8 \times 11 \times 3}\right) a^2 b^2 c^2 = \left(\frac{176}{264}\right) a^2 b^2 c^2 = \frac{2}{3} a^2 b^2 c^2\)

c) \(= \left(\frac{3}{8}\right) \times \left(\frac{-4}{7}\right) \times \left(\frac{-7}{11}\right) \times x^{3+1} \times y^{1+2} \times z^{1+2}\)
\(= \left(\frac{3 \times (-4) \times (-7)}{8 \times 7 \times 11}\right) x^4 y^3 z^3 = \frac{84}{616} x^4 y^3 z^3 = \frac{3}{22} x^4 y^3 z^3\)

d) \(= \left(\frac{5}{7}\right) \times \left(\frac{5}{7}\right) \times \left(\frac{5}{22}\right) \times a^{1+2} \times b^{1+1+2} \times c^1\)
\(= \left(\frac{125}{1078}\right) a^3 b^4 c\)
In simple words: Multiply all numerical coefficients together, then multiply variables by adding exponents of the same base.

Exam Tip: Cancel out common factors in numerators and denominators to simplify fractions easily.

2. Find the volume of the box whose dimensions are \(3xy^2\), \(\frac{1}{7}x^3\) and \(\frac{4}{5}x^2y\).

Answer:
Volume = \(\text{Length} \times \text{Breadth} \times \text{Height}\)
\(= (3xy^2) \times \left(\frac{1}{7}x^3\right) \times \left(\frac{4}{5}x^2y\right)\)
\(= \left(3 \times \frac{1}{7} \times \frac{4}{5}\right) \times (x^{1+3+2}) \times (y^{2+1})\)
\(= \frac{12}{35} x^6 y^3\)
In simple words: Multiply all three dimensions together to find the volume.

Exam Tip: Volume units are always cubic units; keep track of total powers of each variable.

3. Find the area of a field whose length is \(3x^2y + y^2\) and breadth is \(\frac{5}{8}x^2\).

Answer:
Area = \(\text{Length} \times \text{Breadth}\)
\(= (3x^2y + y^2) \times \left(\frac{5}{8}x^2\right)\)
\(= \left(3x^2y \times \frac{5}{8}x^2\right) + \left(y^2 \times \frac{5}{8}x^2\right)\)
\(= \frac{15}{8} x^4 y + \frac{5}{8} x^2 y^2\)
In simple words: Multiply the monomial breadth across each term of the binomial length using the distributive property.

Exam Tip: Distribute the monomial to every term inside the binomial parentheses.

Find the products of the following expressions.

1. \((2x - 1)(x + 2) =\)

Answer:
\(= 2x(x + 2) - 1(x + 2)\)
\(= 2x^2 + 4x - x - 2\)
\(= 2x^2 + 3x - 2\)
In simple words: Multiply each term of the first binomial by each term of the second binomial (FOIL method).

Exam Tip: Combine middle like terms (\(4x - x\)) to simplify the expression.

2. \((p^2 + q^2)(p + q) =\)

Answer:
\(= p^2(p + q) + q^2(p + q)\)
\(= p^3 + p^2q + p q^2 + q^3\)
In simple words: Expand by distributing each term of the first bracket into the second bracket.

Exam Tip: Ensure all four distinct products are written down before concluding.

3. \(\left(\frac{2}{3}x + y\right)(x^2 - y^2) =\)

Answer:
\(= \frac{2}{3}x(x^2 - y^2) + y(x^2 - y^2)\)
\(= \frac{2}{3}x^3 - \frac{2}{3}xy^2 + yx^2 - y^3\)
In simple words: Distribute both terms of the binomial across the binomial.

Exam Tip: Keep fractional coefficients exact throughout multiplication.

4. \((2x - 1)(x^2 + x + 1) =\)

Answer:
\(= 2x(x^2 + x + 1) - 1(x^2 + x + 1)\)
\(= 2x^3 + 2x^2 + 2x - x^2 - x - 1\)
\(= 2x^3 + (2x^2 - x^2) + (2x - x) - 1\)
\(= 2x^3 + x^2 + x - 1\)
In simple words: Multiply every term in the trinomial by each term in the binomial.

Exam Tip: Align like terms carefully when simplifying polynomial products.

5. \((3a + 1)\left(\frac{a}{5} - a + 1\right) =\)

Answer:
First simplify the trinomial: \(\frac{a}{5} - a + 1 = \frac{a - 5a + 5}{5} = \frac{-4a + 5}{5}\)
Now multiply by \((3a + 1)\):
\(= (3a + 1)\left(\frac{-4a + 5}{5}\right)\)
\(= \frac{1}{5} [3a(-4a + 5) + 1(-4a + 5)]\)
\(= \frac{1}{5} [-12a^2 + 15a - 4a + 5]\)
\(= \frac{1}{5} [-12a^2 + 11a + 5]\)
\(= \frac{-12}{5}a^2 + \frac{11}{5}a + 1\)
In simple words: Simplify like terms inside the second bracket first, then multiply.

Exam Tip: Combining inner like terms first simplifies the multiplication process.

6. \((x + xy)\left(\frac{x^2}{2} + xy + y^2\right) =\)

Answer:
\(= x\left(\frac{x^2}{2} + xy + y^2\right) + xy\left(\frac{x^2}{2} + xy + y^2\right)\)
\(= \frac{x^3}{2} + x^2y + xy^2 + \frac{x^3y}{2} + x^2y^2 + xy^3\)
In simple words: Expand each term of the binomial across the trinomial.

Exam Tip: No like terms can be combined here, so write all distinct product terms.

1. Write the squares of the following binomials.
a) \((2x + 5)\)
b) \((3x - 5)\)
c) \(\left(x - \frac{1}{x}\right)\)
d) \(\left(2x + \frac{3}{y}\right)\)

Answer:
a) \((2x + 5)^2 = (2x)^2 + 2(2x)(5) + (5)^2 = 4x^2 + 20x + 25\)
b) \((3x - 5)^2 = (3x)^2 - 2(3x)(5) + (5)^2 = 9x^2 - 30x + 25\)
c) \(\left(x - \frac{1}{x}\right)^2 = (x)^2 - 2(x)\left(\frac{1}{x}\right) + \left(\frac{1}{x}\right)^2 = x^2 - 2 + \frac{1}{x^2}\)
d) \(\left(2x + \frac{3}{y}\right)^2 = (2x)^2 + 2(2x)\left(\frac{3}{y}\right) + \left(\frac{3}{y}\right)^2 = 4x^2 + \frac{12x}{y} + \frac{9}{y^2}\)
In simple words: Use the standard algebraic identities \((a \pm b)^2 = a^2 \pm 2ab + b^2\).

Exam Tip: Remember the middle term is twice the product of both terms.

2. Find the following products using an identity.
a) \((2a + b)(2a - b) =\)
b) \(\left(3x + \frac{1}{y}\right)\left(3x - \frac{1}{y}\right) =\)

Answer:
a) Using identity \((a + b)(a - b) = a^2 - b^2\):
\((2a + b)(2a - b) = (2a)^2 - (b)^2 = 4a^2 - b^2\)

b) Using identity \((a + b)(a - b) = a^2 - b^2\):
\(\left(3x + \frac{1}{y}\right)\left(3x - \frac{1}{y}\right) = (3x)^2 - \left(\frac{1}{y}\right)^2 = 9x^2 - \frac{1}{y^2}\)
In simple words: The product of the sum and difference of two terms is the difference of their squares.

Exam Tip: Check that the first and second terms in both brackets are identical except for the sign.

3. Evaluate the following by using the formulae for \((a + b)^2\) and \((a - b)^2\).
a) \((101)^2 =\)
b) \((99)^2 =\)
c) \((198)^2 =\)

Answer:
a) \((101)^2 = (100 + 1)^2 = (100)^2 + 2(100)(1) + (1)^2 = 10000 + 200 + 1 = 10201\)
b) \((99)^2 = (100 - 1)^2 = (100)^2 - 2(100)(1) + (1)^2 = 10000 - 200 + 1 = 9801\)
c) \((198)^2 = (200 - 2)^2 = (200)^2 - 2(200)(2) + (2)^2 = 40000 - 800 + 4 = 39204\)
In simple words: Split numbers into a round base (like 100 or 200) plus or minus a small number.

Exam Tip: Splitting into powers of 10 makes mental calculation much faster and error-free.

1. Evaluate the following using suitable identities.
a) \(82^2 - 18^2 =\)
b) \(95 \times 105 =\)
c) \(196 \times 204 =\)
d) \(\frac{65^2 - 20^2}{85} =\)

Answer:
a) Using \(a^2 - b^2 = (a + b)(a - b)\):
\(82^2 - 18^2 = (82 + 18)(82 - 18) = (100)(64) = 6400\)

b) Using \((a - b)(a + b) = a^2 - b^2\):
\(95 \times 105 = (100 - 5)(100 + 5) = 100^2 - 5^2 = 10000 - 25 = 9975\)

c) Using \((a - b)(a + b) = a^2 - b^2\):
\(196 \times 204 = (200 - 4)(200 + 4) = 200^2 - 4^2 = 40000 - 16 = 39984\)

d) Using \(a^2 - b^2 = (a + b)(a - b)\) in numerator:
\(\frac{(65 + 20)(65 - 20)}{85} = \frac{(85)(45)}{85} = 45\)
In simple words: Apply standard algebraic identities to simplify difficult multiplications and subtractions.

Exam Tip: Look for the average of the two numbers to find the base value.

2. Find the value of \(x^2 + \frac{1}{x^2}\) if \(x + \frac{1}{x} = 3\).

Answer:
Given \(x + \frac{1}{x} = 3\)
Squaring both sides:
\(\left(x + \frac{1}{x}\right)^2 = 3^2\)
\(x^2 + 2(x)\left(\frac{1}{x}\right) + \frac{1}{x^2} = 9\)
\(x^2 + 2 + \frac{1}{x^2} = 9\)
\(x^2 + \frac{1}{x^2} = 9 - 2 = 7\)
In simple words: Square the given equation on both sides and simplify.

Exam Tip: The middle term \(2 \cdot x \cdot \frac{1}{x}\) simplifies simply to 2.

3. Find the value of \(x^2 + \frac{1}{x^2}\) if \(x - \frac{1}{x} = 4\).

Answer:
Given \(x - \frac{1}{x} = 4\)
Squaring both sides:
\(\left(x - \frac{1}{x}\right)^2 = 4^2\)
\(x^2 - 2(x)\left(\frac{1}{x}\right) + \frac{1}{x^2} = 16\)
\(x^2 - 2 + \frac{1}{x^2} = 16\)
\(x^2 + \frac{1}{x^2} = 16 + 2 = 18\)
In simple words: Square both sides of the subtraction identity to find the sum of squares.

Exam Tip: Notice that minus in the linear term results in adding 2 when moved to the right side.

4. If \(x + y = 4\) and \(xy = 3\), find the value of \(x^2 + y^2\).

Answer:
We know that:
\((x + y)^2 = x^2 + y^2 + 2xy\)
Substitute given values \(x + y = 4\) and \(xy = 3\):
\(4^2 = x^2 + y^2 + 2(3)\)
\(16 = x^2 + y^2 + 6\)
\(x^2 + y^2 = 16 - 6 = 10\)
In simple words: Use the expansion of \((x + y)^2\) to isolate \(x^2 + y^2\).

Exam Tip: Memorize basic expansion formulas for quick substitution.

5. If \(x^2 + \frac{1}{x^2} = 9\), find the value of \(x^4 + \frac{1}{x^4}\).

Answer:
Given \(x^2 + \frac{1}{x^2} = 9\)
Squaring both sides:
\(\left(x^2 + \frac{1}{x^2}\right)^2 = 9^2\)
\((x^2)^2 + 2(x^2)\left(\frac{1}{x^2}\right) + \left(\frac{1}{x^2}\right)^2 = 81\)
\(x^4 + 2 + \frac{1}{x^4} = 81\)
\(x^4 + \frac{1}{x^4} = 81 - 2 = 79\)
In simple words: Square the given equation again to jump from power 2 to power 4.

Exam Tip: Repeated squaring is a standard technique for higher power problems.

1. Evaluate the following using suitable identities.
a) \((196)^2 =\)
b) \(52 \times 48 =\)
c) \((205)^2 =\)
d) \((108)^2 =\)

Answer:
a) \((200 - 4)^2 = 200^2 - 2(200)(4) + 4^2 = 40000 - 1600 + 16 = 38416\)
b) \((50 + 2)(50 - 2) = 50^2 - 2^2 = 2500 - 4 = 2496\)
c) \((200 + 5)^2 = 200^2 + 2(200)(5) + 5^2 = 40000 + 2000 + 25 = 42025\)
d) \((100 + 8)^2 = 100^2 + 2(100)(8) + 8^2 = 10000 + 1600 + 64 = 11664\)
In simple words: Express numbers near multiples of 100 as sums or differences.

Exam Tip: Double check signs when expanding binomial squares.

2. If \(x + \frac{1}{x} = 5\), find the value of \(x^2 + \frac{1}{x^2}\) and \(x^4 + \frac{1}{x^4}\).

Answer:
Given \(x + \frac{1}{x} = 5\)
Squaring both sides:
\(\left(x + \frac{1}{x}\right)^2 = 5^2\)
\(x^2 + 2 + \frac{1}{x^2} = 25\)
\(x^2 + \frac{1}{x^2} = 23\)
Now, square again to find \(x^4 + \frac{1}{x^4}\):
\(\left(x^2 + \frac{1}{x^2}\right)^2 = 23^2\)
\(x^4 + 2 + \frac{1}{x^4} = 529\)
\(x^4 + \frac{1}{x^4} = 529 - 2 = 527\)
In simple words: Square once for power 2, then square the result to get power 4.

Exam Tip: Remember that \(23^2 = 529\).

3. If \(x^2 + \frac{1}{x^2} = 7\) find the value of \(x + \frac{1}{x}, \; x > 0\).

Answer:
We know that:
\(\left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2\)
Substitute \(x^2 + \frac{1}{x^2} = 7\):
\(\left(x + \frac{1}{x}\right)^2 = 7 + 2 = 9\)
Taking square root on both sides (since \(x > 0\)):
\(x + \frac{1}{x} = \sqrt{9} = 3\)
In simple words: Add 2 to the given value and take the square root.

Exam Tip: Condition \(x > 0\) ensures we take only the positive square root.

4. If \(x + y = 5\) and \(xy = 6\). Find the value of \(x^2 + y^2\) and \(x - y\).

Answer:
Part 1: Find \(x^2 + y^2\)
\((x + y)^2 = x^2 + y^2 + 2xy\)
\(5^2 = x^2 + y^2 + 2(6)\)
\(25 = x^2 + y^2 + 12\)
\(x^2 + y^2 = 13\)
Part 2: Find \(x - y\)
We know that \((x - y)^2 = x^2 + y^2 - 2xy\)
\((x - y)^2 = 13 - 2(6) = 13 - 12 = 1\)
\(x - y = \pm 1\)
In simple words: Use the relation between \((x+y)^2\) and \((x-y)^2\) via \(x^2 + y^2\).

Exam Tip: Keep track of both positive and negative roots where applicable.

1. Find the following products:
a) \((x + 2)(3x + 1)(x - 3) =\)
b) \((5x + 1)(2x - 3)(x + 5) =\)
c) \((2x + y)(x - y)(x + y) =\)
d) \((3a + b)(2a + b)(a - b) =\)

Answer:
a) First multiply \((x + 2)(3x + 1) = 3x^2 + x + 6x + 2 = 3x^2 + 7x + 2\)
Now multiply by \((x - 3)\):
\((3x^2 + 7x + 2)(x - 3) = 3x^2(x - 3) + 7x(x - 3) + 2(x - 3)\)
\(= 3x^3 - 9x^2 + 7x^2 - 21x + 2x - 6\)
\(= 3x^3 - 2x^2 - 19x - 6\)

b) First multiply \((5x + 1)(2x - 3) = 10x^2 - 15x + 2x - 3 = 10x^2 - 13x - 3\)
Now multiply by \((x + 5)\):
\((10x^2 - 13x - 3)(x + 5) = 10x^3 + 50x^2 - 13x^2 - 65x - 3x - 15\)
\(= 10x^3 + 37x^2 - 68x - 15\)

c) Notice \((x - y)(x + y) = x^2 - y^2\). Now multiply by \((2x + y)\):
\((2x + y)(x^2 - y^2) = 2x(x^2 - y^2) + y(x^2 - y^2)\)
\(= 2x^3 - 2xy^2 + yx^2 - y^3\)

d) First multiply \((2a + b)(a - b) = 2a^2 - 2ab + ab - b^2 = 2a^2 - ab - b^2\)
Now multiply by \((3a + b)\):
\((3a + b)(2a^2 - ab - b^2) = 3a(2a^2 - ab - b^2) + b(2a^2 - ab - b^2)\)
\(= 6a^3 - 3a^2b - 3ab^2 + 2a^2b - ab^2 - b^3\)
\(= 6a^3 - a^2b - 4ab^2 - b^3\)
In simple words: Multiply two binomials first, then multiply the resulting polynomial by the third factor.

Exam Tip: Look for standard identities like \((x-y)(x+y)\) to simplify triple products faster.

2. Factorise the following using suitable identities.
a) \(49x^2 - 9^2 =\)
b) \(16x^2 - 25 =\)
c) \(x^2 - 2x + 1 =\)
d) \(9x^2 - 6x + 1 =\)
e) \(x^2 - (y + z)^2 =\)

Answer:
a) \(49x^2 - 81 = (7x)^2 - 9^2 = (7x + 9)(7x - 9)\)
b) \(16x^2 - 25 = (4x)^2 - 5^2 = (4x + 5)(4x - 5)\)
c) \(x^2 - 2x + 1 = (x - 1)^2 = (x - 1)(x - 1)\)
d) \(9x^2 - 6x + 1 = (3x)^2 - 2(3x)(1) + 1^2 = (3x - 1)^2\)
e) \(x^2 - (y + z)^2 = [x + (y + z)][x - (y + z)] = (x + y + z)(x - y - z)\)
In simple words: Recognize squares of binomials and difference of squares to factorise.

Exam Tip: Be careful with signs when factorising the difference of two compound terms as in part (e).