Algebraic Expressions and Operations
Algebraic Expressions and Operations
Algebra is worth 6 to 8 questions in the 40-mark Part-III maths paper, second only to geometry. This first topic establishes the vocabulary — term, coefficient, degree — that every later question assumes. Get the definitions exact and the rest of the chapter becomes mechanical.
The Vocabulary of an Expression
In the expression 5x²y − 3xy + 7, the parts separated by plus or minus signs are terms. There are three of them.
| Word | Meaning | In 5x²y |
|---|---|---|
| Term | A part separated by + or − | 5x²y |
| Coefficient | The numerical factor | 5 |
| Variable | The letter | x and y |
| Constant | Term with no variable | 7 |
| Degree of a term | Sum of the exponents | 2 + 1 = 3 |
| Degree of expression | Highest degree among terms | 3 |
Expressions are named by their number of terms: monomial for one, binomial for two, trinomial for three, and polynomial in general.
✗ The degree of 5x²y is 2 | ✓ Degree is the sum of the exponents, so it is 3
Like and Unlike Terms
Only like terms can be added or subtracted. Like terms have exactly the same variables raised to exactly the same powers; their coefficients may differ.
- 3xy and −7xy are like terms — they combine to −4xy.
- 3x²y and 3xy² are not like terms; they cannot be combined at all.
- Multiplication, unlike addition, works on any terms: 3x²y × 3xy² = 9x³y³.
This asymmetry — addition restricted, multiplication free — is the point students miss and papers test.
Multiplication of Expressions
(a + b)(c + d) = ac + ad + bc + bd
Every term of the first bracket multiplies every term of the second. For (x + 3)(x + 5): x² + 5x + 3x + 15 = x² + 8x + 15.
Notice the pattern in the answer: the middle coefficient is the sum of 3 and 5, the constant is their product. Recognising this in reverse is what makes factorisation quick in the third topic of this chapter.
Substitution and Evaluation
To evaluate an expression, replace each variable with its value and follow BODMAS carefully — especially with negative values.
If x = −2, then x² = 4 but −x² = −4, because the exponent applies to x alone, not to the minus sign. The square of a negative is positive, but the negative of a square stays negative. That single distinction accounts for a large share of lost marks in algebra.
✗ If x = −2 then −x² = 4 | ✓ −x² means −(x²) = −4
समान पद ही जोड़े जा सकते हैं, पर गुणा किसी भी पद में किया जा सकता है।
TRE pointer: Part-III mathematics carries 40 questions against just 8 in Part-II, so algebra here is examined at proper depth, not as arithmetic in disguise. Definition questions — degree of a term, number of terms, identifying like terms — are common because they can be set unambiguously. The −x² versus (−x)² distinction is the classic trap and both values will sit among the five options. With −1/3 for a wrong answer, substitute on paper rather than in your head.
60-Second Recap
- Terms are separated by + or −; the coefficient is the numerical factor.
- Degree of a term is the sum of its exponents; degree of an expression is the highest such sum.
- Monomial, binomial, trinomial by number of terms.
- Only like terms can be added or subtracted; any terms can be multiplied.
- (a+b)(c+d) expands to four products; the middle coefficient is a sum, the constant a product.
- −x² is not the same as (−x)².
Frequently Asked Questions
What is the degree of an algebraic expression?
It is the highest degree among its terms, where the degree of a term is the sum of the exponents of its variables. In 5x squared y minus 3xy plus 7, the first term has degree three and the second degree two, so the expression has degree three.
Can 3x squared y and 3xy squared be added?
No. Like terms must have the same variables raised to exactly the same powers, and here the powers are swapped. They cannot be combined by addition at all. They can, however, be multiplied, giving nine x cubed y cubed, since multiplication has no such restriction.
What is the difference between minus x squared and minus x, all squared?
In minus x squared the exponent applies only to x, so with x equal to minus two the value is minus four. In minus x, all squared, the whole quantity is squared, giving positive four. Both values are placed among the options deliberately.
How much algebra comes in BPSC TRE 4.0 Part-III?
Algebra typically accounts for six to eight of the forty mathematics questions, covering expressions, identities, factorisation, linear equations and graphs. Since Part-III marks count fully towards merit, this is one of the highest-value blocks in the entire paper.