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Algebraic Expressions & Identities

Multiplying Polynomials

Addition and Subtraction of Algebraic Expressions

To add or subtract algebraic expressions:

  • Group like terms together
  • Add or subtract the coefficients of like terms
  • Keep unlike terms as they are

Example: \((3x^2 + 2x - 5) + (4x^2 - 3x + 7) = (3x^2 + 4x^2) + (2x - 3x) + (-5 + 7) = 7x^2 - x + 2\)

Multiplication of Algebraic Expressions

TypeRuleExample
Monomial × MonomialMultiply coefficients, add exponents of like variables\((3x^2)(4x^3) = 12x^{5}\)
Monomial × PolynomialDistributive property: Multiply monomial by each term\(2x(x + 3) = 2x^2 + 6x\)
Binomial × BinomialFOIL method or distributive property\((x + 2)(x + 3) = x^2 + 5x + 6\)
Polynomial × PolynomialMultiply each term of first by each term of second\((x + 1)(x^2 + 2x + 3) = x^3 + 3x^2 + 5x + 3\)

Division of Algebraic Expressions

  • Monomial ÷ Monomial: Divide coefficients, subtract exponents of like variables
  • Example: \(\frac{15x^5}{3x^2} = 5x^{3}\)
  • Polynomial ÷ Monomial: Divide each term by the monomial
  • Example: \(\frac{6x^3 + 9x^2}{3x} = 2x^2 + 3x\)
  • Polynomial ÷ Binomial: Use long division method

Simplification of Algebraic Expressions

Simplification means combining like terms, applying distributive property, and reducing to the simplest form without changing the value.

Factorization Basics

Factorization is the process of writing an expression as a product of its factors (the reverse of multiplication).

Methods of Factorization:

  • Taking out common factors: \(4x^2 + 6x = 2x(2x + 3)\)
  • Grouping: \(ax + ay + bx + by = a(x + y) + b(x + y) = (x + y)(a + b)\)
  • Using identities: \(x^2 + 6x + 9 = (x + 3)^2\)
Adding & Subtracting Algebraic ExpressionsStep 1: Write expressions(3x + 2y − 4) + (2x − y + 6)Step 2: Group like terms(3x + 2x) + (2y − y) + (−4 + 6)Step 3: Simplify5x + y + 2Multiplication: Use FOIL or distributive law — (a+b)(c+d) = ac+ad+bc+bdDivision: Divide each term by the divisor separately
Example 1: Expand 2x(x + 3).
2x² + 6x.
Example 2: Expand (x + 2)(x + 3).
x² + 3x + 2x + 6 = x² + 5x + 6.
Example 3: Example 1: Add: \((5a^2 - 3ab + 2b^2)\) and \((-2a^2 + 7ab - b^2)\)
- Step 1: Group like terms together - Step 2: \(a^2\) terms: \(5a^2 + (-2a^2) = 3a^2\) - Step 3: \(ab\) terms: \((-3ab) + 7ab = 4ab\) - Step 4: \(b^2\) terms: \(2b^2 + (-b^2) = b^2\) - Step 5: Combine all: \(3a^2 + 4ab + b^2\)
Example 4: Example 2: Multiply: \((2x - 3)(x^2 + 4x - 5)\)
- Step 1: Multiply each term of first expression with each term of second - Step 2: \(2x \times x^2 = 2x^3\) - Step 3: \(2x \times 4x = 8x^2\) - Step 4: \(2x \times (-5) = -10x\) - Step 5: \((-3) \times x^2 = -3x^2\) - Step 6: \((-3) \times 4x = -12x\) - Step 7: \((-3) \times (-5) = 15\) - Step 8: Combine like terms: \(2x^3 + (8x^2 - 3x^2) = 2x^3 + 5x^2\) - Step 9: \((-10x - 12x) = -22x\), then \(+15\) - Step 10: Final expression: \(2x^3 + 5x^2 - 22x + 15\)
Example 5: Example 3: Factorize completely: \(4x^2y + 8xy^2 + 12xy\)
- Step 1: Find the greatest common factor (GCF) of all terms - Step 2: Numbers: \(4, 8, 12\) have GCF = \(4\) - Step 3: Variables: \(x^2y\), \(xy^2\), \(xy\) have GCF = \(xy\) - Step 4: Overall GCF = \(4xy\) - Step 5: Divide each term by \(4xy\): - \(4x^2y ÷ 4xy = x\) - \(8xy^2 ÷ 4xy = 2y\) - \(12xy ÷ 4xy = 3\) - Step 6: Write as product: \(4xy(x + 2y + 3)\)
Quick recap
  • Monomial × polynomial: multiply each term.
  • Binomial × binomial: multiply all pairs, then collect like terms.
  • Addition/Subtraction: Only like terms can be added or subtracted
  • Multiplication: Multiply coefficients, add exponents of same variables
  • FOIL method: First, Outer, Inner, Last for multiplying two binomials
  • Division: Subtract exponents for monomials; use long division for polynomials
  • Simplification: Combine like terms and remove brackets using distributive property
  • Factorization: Reverse of multiplication - express as product of factors
  • Always check if the result can be simplified further
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✓ Quick check
Expand: 3(x + 4).
3 × x + 3 × 4 = 3x + 12.
Expand: (x + 1)(x + 2).
x² + 2x + x + 2 = x² + 3x + 2.

Standard Identities

What are Algebraic Identities?

Algebraic identities are equations that are true for all values of the variables. They help us simplify complex algebraic expressions and perform calculations mentally.

The Three Standard Identities:

IdentityFormulaName
Identity I\((a + b)^2 = a^2 + 2ab + b^2\)Square of a Sum
Identity II\((a - b)^2 = a^2 - 2ab + b^2\)Square of a Difference
Identity III\((a + b)(a - b) = a^2 - b^2\)Difference of Squares

Verification of Identities:

For Identity I: \((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2\)

Applications of Identities:

  • Expanding expressions: \((3x + 5)^2 = (3x)^2 + 2(3x)(5) + 5^2 = 9x^2 + 30x + 25\)
  • Factorizing expressions: \(x^2 - 9 = (x)^2 - (3)^2 = (x + 3)(x - 3)\)
  • Simplifying calculations mentally: \(102^2 = (100 + 2)^2 = 100^2 + 2(100)(2) + 2^2 = 10000 + 400 + 4 = 10404\)

Mental Mathematics Using Identities:

These identities make it easy to calculate squares and products of large numbers mentally:

CalculationUsing IdentityMental Math
\(35^2\)\((30 + 5)^2 = 900 + 300 + 25 = 1225\)Add 300 to 900 to get 1200, plus 25
\(28 \times 32\)\((30 - 2)(30 + 2) = 900 - 4 = 896\)Square of middle number minus square of difference
\(99^2\)\((100 - 1)^2 = 10000 - 200 + 1 = 9801\)Subtract 200 from 10000, add 1

More Useful Identities:

\((x + a)(x + b) = x^2 + (a + b)x + ab\)

\((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)

Standard Identity: (a + b)² = a² + 2ab + b²a²b²ababababKey Identities(a+b)² = a²+2ab+b²(a−b)² = a²−2ab+b²a²−b² = (a+b)(a−b)(a+b+c)² = a²+b²+c² +2ab+2bc+2ca
Example 1: Expand (x + 3)².
x² + 2(x)(3) + 9 = x² + 6x + 9.
Example 2: Expand (x − 2)².
x² − 4x + 4.
Example 3: Example 1: Using identity, find the value of \((5x + 3y)^2\)
- Step 1: Identify \(a = 5x\) and \(b = 3y\) - Step 2: Use identity \((a + b)^2 = a^2 + 2ab + b^2\) - Step 3: \(a^2 = (5x)^2 = 25x^2\) - Step 4: \(2ab = 2 \times 5x \times 3y = 30xy\) - Step 5: \(b^2 = (3y)^2 = 9y^2\) - Step 6: Combine: \(25x^2 + 30xy + 9y^2\)
Example 4: Example 2: Calculate \(98^2\) mentally using algebraic identity.
- Step 1: Write \(98 = 100 - 2\) - Step 2: Use \((a - b)^2 = a^2 - 2ab + b^2\) with \(a = 100\), \(b = 2\) - Step 3: \(a^2 = 10000\) - Step 4: \(2ab = 2 \times 100 \times 2 = 400\) - Step 5: \(b^2 = 4\) - Step 6: \(10000 - 400 + 4 = 9600 + 4 = 9604\)
Example 5: Example 3: Simplify: \((2x + 5)^2 - (2x - 5)^2\) using identities.
- Step 1: Identify \(a = 2x\), \(b = 5\) - Step 2: \((a + b)^2 = a^2 + 2ab + b^2 = 4x^2 + 20x + 25\) - Step 3: \((a - b)^2 = a^2 - 2ab + b^2 = 4x^2 - 20x + 25\) - Step 4: Subtract: \((4x^2 + 20x + 25) - (4x^2 - 20x + 25)\) - Step 5: Simplify: \(4x^2 + 20x + 25 - 4x^2 + 20x - 25\) - Step 6: Combine: \((4x^2 - 4x^2) + (20x + 20x) + (25 - 25) = 40x\) - Step 7: Alternatively, use identity directly: \((a+b)^2 - (a-b)^2 = 4ab = 4(2x)(5) = 40x\)
Example 6: Example 4 (Application): The area of a square is \(9x^2 + 12x + 4\). Find the length of its side.
- Step 1: Recognize that area of square = (side)\(^2\) - Step 2: So side\(^2 = 9x^2 + 12x + 4\) - Step 3: Check if it matches \((a + b)^2 = a^2 + 2ab + b^2\) - Step 4: \(a^2 = 9x^2\) → \(a = 3x\) - Step 5: \(b^2 = 4\) → \(b = 2\) - Step 6: Check \(2ab = 2 \times 3x \times 2 = 12x\) ✓ - Step 7: Therefore, \(9x^2 + 12x + 4 = (3x + 2)^2\) - Step 8: Side = \(3x + 2\)
Quick recap
  • (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b².
  • (a + b)(a − b) = a² − b².
  • Identity I: \((a + b)^2 = a^2 + 2ab + b^2\) (Square of sum)
  • Identity II: \((a - b)^2 = a^2 - 2ab + b^2\) (Square of difference)
  • Identity III: \((a + b)(a - b) = a^2 - b^2\) (Difference of squares)
  • Identities are true for all values of variables (unlike equations which are true only for specific values)
  • Use identities to expand expressions and factorize expressions quickly
  • Mental math: \(105^2 = (100+5)^2 = 10000 + 1000 + 25 = 11025\)
  • For products like \(28 \times 32\), write as \((30-2)(30+2) = 900 - 4 = 896\)
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✓ Quick check
(a + b)(a − b) equals ___ ?
The difference of squares: a² − b².
Expand (x + 5)².
x² + 2(5)x + 25 = x² + 10x + 25.

(x + a)(x + b) & Applying Identities

Another identity is (x + a)(x + b) = x² + (a + b)x + ab. Identities also speed up arithmetic: 101 × 99 = (100 + 1)(100 − 1) = 10000 − 1 = 9999.

Example 1: Expand (x + 2)(x + 5).
x² + 7x + 10.
Example 2: Compute 102² using an identity.
(100 + 2)² = 10000 + 400 + 4 = 10404.
Quick recap
  • (x + a)(x + b) = x² + (a + b)x + ab.
  • Use identities to compute products quickly.
✓ Quick check
Expand (x + 3)(x + 4).
x² + (3 + 4)x + 12 = x² + 7x + 12.
Compute 102² using an identity.
(100 + 2)² = 10000 + 400 + 4 = 10404.
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