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Algebraic expressions and formulas: Class 9 Mathematics
Constants, variables and terms, polynomials and their degree, addition, subtraction and multiplication of expressions, the standard algebraic identities for squares and cubes, and evaluating expressions using these formulas.
Key points
In algebra, letters stand for numbers. A constant has a fixed value, such as 7 or −3. A variable can take different values, such as x or y. A term is a number, a variable, or a product of these, for example 5x² or −3xy. Terms are separated by plus or minus signs. The number multiplying the variable part is the coefficient, so the coefficient of 5x² is 5.
An algebraic expression is a combination of constants and variables joined by operations. A polynomial in x has terms of the form axⁿ where n is a whole number. The degree of a polynomial in one variable is the highest power of x. So 4x³ − 2x + 1 has degree 3. For a term with two variables, such as 3x²y, the degree is the sum of the powers, here 3.
Types of expressions by number of terms:
- Monomial: one term, for example 6x².
- Binomial: two terms, for example x + 5.
- Trinomial: three terms, for example x² + 3x + 2.
- Polynomial: any number of terms, each with a whole number power.
Like terms have the same variables with the same powers, such as 3x²y and −5x²y. Only like terms can be added or subtracted, by adding their coefficients. So 3x²y − 5x²y = −2x²y. Unlike terms such as 3x and 3x² cannot be combined.
To add polynomials, write them in a row or column with like terms together and add the coefficients. To subtract, change the sign of every term in the second polynomial and then add. A bracket with a minus sign in front changes every sign inside it: −(2x − 3) = −2x + 3.
To multiply, use the law aᵐ × aⁿ = aᵐ⁺ⁿ for powers and the distributive law for brackets. For a monomial times a polynomial, multiply every term inside the bracket. For two binomials, multiply each term of the first by each term of the second, then collect like terms: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15.
Division of a polynomial by a monomial divides each term separately, using aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example (6x³ + 9x²) ÷ 3x = 2x² + 3x. Long division is used when dividing by a polynomial with more than one term.
Algebraic formulas (identities) for squares:
- (a + b)² = a² + 2ab + b².
- (a − b)² = a² − 2ab + b².
- a² − b² = (a + b)(a − b).
- (a + b)² − (a − b)² = 4ab and (a + b)² + (a − b)² = 2(a² + b²).
- (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
An identity is true for all values of the letters. These formulas can be checked by expansion. For (a + b)², write (a + b)(a + b) = a² + ab + ba + b² = a² + 2ab + b². Geometrically, a square of side a + b is made of a square a², a square b² and two rectangles each of area ab.
Algebraic formulas for cubes:
- (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).
Formulas help with mental arithmetic. To find 103², write (100 + 3)² = 10000 + 600 + 9 = 10609. To find 98 × 102, write (100 − 2)(100 + 2) = 100² − 2² = 9996. They are also used to find a² + b² when a + b and ab are known, and x² + 1/x² when x + 1/x is known.
To evaluate an expression, replace each letter by its value and follow the order of operations: brackets, powers, multiplication and division, then addition and subtraction. Take care with negative values, since (−3)² = 9 but −3² = −9. Put negative numbers in brackets when substituting.
Study tip: Memorise the square and cube formulas exactly, with their signs, and write the formula before substituting. In expansion questions always compare the number of terms you expect: (a + b + c)² must give six terms. For numerical-trick questions, choose a base such as 100 or 50 so the sum or difference is small.
Definitions and flashcards
Say the meaning first, then open the card to check.
Variable
A letter that can stand for different numbers.
Coefficient
The number multiplying the variable part of a term.
Polynomial
An expression whose terms are numbers times whole-number powers of variables.
Degree
The highest power of the variable in a polynomial.
Like terms
Terms with the same variables raised to the same powers.
Binomial
An expression with exactly two terms.
Identity
An equation that is true for all values of its letters.
Trinomial
An expression with exactly three terms.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Square of a sum | (a + b)² = a² + 2ab + b² | Expand a binomial sum squared. |
| Square of a difference | (a − b)² = a² − 2ab + b² | Expand a binomial difference squared. |
| Difference of squares | a² − b² = (a + b)(a − b) | Product of sum and difference. |
| Cube of a sum | (a + b)³ = a³ + 3a²b + 3ab² + b³ | Expand a sum cubed. |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) | Factor form of a sum of cubes. |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) | Factor form of a difference of cubes. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
Define a polynomial and state the degree of 5x⁴ − 3x² + 7. (2 marks)
A polynomial is an algebraic expression in which every term is a constant times a whole-number power of the variable. The highest power of x in 5x⁴ − 3x² + 7 is 4, so its degree is 4.
What are like terms? Give an example. (2 marks)
Like terms have the same variables with the same powers. Only their coefficients differ. For example 4ab² and −7ab² are like terms, but 4ab² and 4a²b are not.
Expand (2x − 3y)². (2 marks)
Use (a − b)² = a² − 2ab + b² with a = 2x and b = 3y. a² = 4x², 2ab = 2 × 2x × 3y = 12xy, b² = 9y². So (2x − 3y)² = 4x² − 12xy + 9y².
Prove that (a + b)² − (a − b)² = 4ab. (3 marks)
Expand both: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². Subtract: (a² + 2ab + b²) − (a² − 2ab + b²) = a² + 2ab + b² − a² + 2ab − b² = 4ab. Hence the identity is proved.
Find the product (x + 4)(x² − 4x + 16) using a formula. (3 marks)
The second factor has the form a² − ab + b² with a = x and b = 4, since a² = x², ab = 4x and b² = 16. So the product is the sum of cubes a³ + b³ = x³ + 4³ = x³ + 64.
Expand (a + b)³ step by step and use it to find 11³. (5 marks)
(a + b)³ = (a + b)(a + b)² = (a + b)(a² + 2ab + b²) = a³ + 2a²b + ab² + a²b + 2ab² + b³ = a³ + 3a²b + 3ab² + b³. For 11³ take a = 10 and b = 1: 1000 + 3 × 100 × 1 + 3 × 10 × 1 + 1 = 1000 + 300 + 30 + 1 = 1331.
Numericals with worked steps
If a + b = 7 and ab = 10, find a² + b².
- Use (a + b)² = a² + 2ab + b².
- So a² + b² = (a + b)² − 2ab.
- = 7² − 2 × 10 = 49 − 20 = 29.
Answer: 29
Find 103² using a formula.
- Write 103 = 100 + 3.
- (100 + 3)² = 100² + 2 × 100 × 3 + 3².
- = 10000 + 600 + 9 = 10609.
Answer: 10609
If x + 1/x = 5, find x² + 1/x² and x³ + 1/x³.
- Square: (x + 1/x)² = x² + 2 + 1/x² = 25.
- So x² + 1/x² = 25 − 2 = 23.
- Cube: (x + 1/x)³ = x³ + 1/x³ + 3(x + 1/x) = 125.
- x³ + 1/x³ = 125 − 3 × 5 = 125 − 15 = 110.
Answer: x² + 1/x² = 23 and x³ + 1/x³ = 110
Evaluate 98 × 102 using a formula.
- 98 = 100 − 2 and 102 = 100 + 2.
- (100 − 2)(100 + 2) = 100² − 2².
- = 10000 − 4 = 9996.
Answer: 9996
If a − b = 3 and ab = 4, find a + b.
- (a + b)² = (a − b)² + 4ab.
- = 9 + 16 = 25.
- Taking the square root, a + b = 5 or a + b = −5.
Answer: a + b = ±5
Common mistakes
- Writing (a + b)² = a² + b² and forgetting the middle term 2ab.
- Adding unlike terms, such as 3x + 2x² = 5x³.
- Not changing every sign when removing a bracket that has a minus sign before it.
- Mixing up (−3)² = 9 with −3² = −9 when substituting negative values.
- Using the wrong sign in the cube formulas, for example writing a³ + b³ = (a + b)(a² + ab + b²).