Algebraic Expressions & Polynomial Operationsबीजीय व्यंजक
Mastery of algebraic expressions, polynomial operations (addition, subtraction, multiplication, and division), term identification, coefficient analysis, and degree determination as prescribed in the SCERT UP Class 6-8 and NCERT curriculum for Super TET Junior Part 2C Mathematics.
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Syllabus Focus & High-Yield Strategy: Mastery of algebraic expressions, polynomial operations (addition, subtraction, multiplication, and division), term identification, coefficient analysis, and degree determination as prescribed in the SCERT UP Class 6-8 and NCERT curriculum for Super TET Junior Part 2C Mathematics.
1. Fundamental Axioms, Definitions & Foundational Properties
L-Q-C-B — Linear (1), Quadratic (2), Cubic (3), Biquadratic (4) degrees in ascending order.2. Formulas, Standard Theorems & Algorithmic Procedures
| Category / Case | Formula / Mathematical Rule | Topper Calculation Shortcut |
|---|---|---|
| Addition / Subtraction | Combine only like terms by adding or subtracting their numerical coefficients. $ax^n \pm bx^n = (a \pm b)x^n$. | Group terms by degree vertically or use column alignment to avoid sign omission. |
| Multiplication | Distributive law: $(ax^m)(bx^n) = abx^{m+n}$. Every term of polynomial A multiplies every term of polynomial B. | Use FOIL for binomials: $(a+b)(c+d) = ac + ad + bc + bd$. |
| Division | Polynomial Long Division: $\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$, where $\deg(R) < \deg(D)$. | Use Synthetic Division or Remainder Theorem when dividing by $(x - c)$. |
3. Word Problem Modeling & Practical Applications
4. Mathematical Edge Cases, Sign Conventions & Common Traps
L-I-K-E — Like terms Keep Identical Literal Exponents.Comprehensive Mathematical Matrix & Method Comparison
| Parameter / Dimension | Method / Concept A | Method / Concept B | Exam Speed & Accuracy Takeaway |
|---|---|---|---|
| Addition / Subtraction | Combine like terms only ($ax^n \pm bx^n$) | Adding unlike terms ($ax^2 + bx$) | Unlike terms cannot be combined; keep them separate. |
| Multiplication | Distributive law over all terms | Multiplying coefficients and adding exponents | Exponent rule: $x^m \times x^n = x^{m+n}$. |
| Division | Polynomial long division | Synthetic division / Remainder theorem | Use Remainder Theorem to find remainders instantly without full division. |
| Degree Determination | Single variable (highest exponent) | Multivariable (max sum of exponents in a term) | Always check all terms in multivariable polynomials. |
1. What is the degree of the polynomial $5x^4y^2 - 3x^6 + 2xy^7 - 9$?
The degree of a multivariable polynomial is the maximum sum of the powers of variables in any single term. Let us find the sum of powers for each term: Term 1 ($5x^4y^2$) has degree $4+2=6$. Term 2 ($-3x^6$) has degree $6$. Term 3 ($2xy^7$) has degree $1+7=8$. Term 4 ($-9$) has degree $0$. The maximum degree among all terms is $8$. Therefore, the correct option is $8$.
2. If $P(x) = 2x^3 - 5x^2 + ax + 3$ is divided by $(x - 2)$ and leaves a remainder of $5$, what is the value of $a$?
By the Remainder Theorem, when $P(x)$ is divided by $(x - 2)$, the remainder is $P(2)$. Given remainder = $5$, so $P(2) = 5$. Substitute $x = 2$ into $P(x)$: $2(2)^3 - 5(2)^2 + a(2) + 3 = 5 \implies 2(8) - 5(4) + 2a + 3 = 5 \implies 16 - 20 + 2a + 3 = 5 \implies -1 + 2a = 5 \implies 2a = 6 \implies a = 3$. Therefore, the correct option is $3$.
3. Simplify the algebraic expression: $3x - [2y - {5x - (4y - 3x)}]$
Inside the braces, 5x - (4y - 3x) = 8x - 4y. The square bracket is 2y - (8x - 4y) = 6y - 8x. Thus the full expression is 3x - (6y - 8x) = 11x - 6y. The answer is Option A.
4. What must be added to $x^3 - 3x^2 + 5x - 7$ to obtain $x^3 + 2x^2 - 4x + 1$?
Let the required expression be $E$. Then $(x^3 - 3x^2 + 5x - 7) + E = x^3 + 2x^2 - 4x + 1$. Therefore, $E = (x^3 + 2x^2 - 4x + 1) - (x^3 - 3x^2 + 5x - 7) = (1 - 1)x^3 + (2 - (-3))x^2 + (-4 - 5)x + (1 - (-7)) = 0x^3 + 5x^2 - 9x + 8 = 5x^2 - 9x + 8$. Therefore, the correct option is $5x^2 - 9x + 8$.
5. If $x + \frac{1}{x} = 5$, what is the value of $x^2 + \frac{1}{x^2}$?
Square both sides of the given equation: $\left(x + \frac{1}{x}\right)^2 = 5^2 \implies x^2 + \frac{1}{x^2} + 2(x)\left(\frac{1}{x}\right) = 25 \implies x^2 + \frac{1}{x^2} + 2 = 25 \implies x^2 + \frac{1}{x^2} = 25 - 2 = 23$. Therefore, the correct option is $23$.
6. Find the product $(2x - 3)(3x^2 + 4x - 5)$
Multiply each term of $(2x - 3)$ with $(3x^2 + 4x - 5)$: $2x(3x^2 + 4x - 5) - 3(3x^2 + 4x - 5) = 6x^3 + 8x^2 - 10x - 9x^2 - 12x + 15$. Group like terms: $6x^3 + (8x^2 - 9x^2) + (-10x - 12x) + 15 = 6x^3 - x^2 - 22x + 15$. Therefore, the correct option is $6x^3 - x^2 - 22x + 15$.
7. Which of the following expressions is a polynomial in one variable $x$?
A polynomial in one variable must have only non-negative integer exponents for the variable $x$. Option 1 has $\sqrt{x} = x^{1/2}$ (fractional exponent). Option 2 has $\frac{1}{x} = x^{-1}$ (negative exponent). Option 4 has two variables ($x$ and $y$). Option 3 ($x^4 - 3x^2 + 7$) has only variable $x$ with integer exponents $4, 2, 0$. Therefore, the correct option is $x^4 - 3x^2 + 7$.
8. If $x - 2$ is a factor of $x^3 - 4x^2 + kx - 6$, what is the value of $k$?
By the Factor Theorem, if $(x - 2)$ is a factor of $P(x)$, then $P(2) = 0$. Substitute $x = 2$: $(2)^3 - 4(2)^2 + k(2) - 6 = 0 \implies 8 - 4(4) + 2k - 6 = 0 \implies 8 - 16 + 2k - 6 = 0 \implies -14 + 2k = 0 \implies 2k = 14 \implies k = 7$. Therefore, the correct option is $7$.
9. What is the coefficient of $x^2$ in the expansion of $(2x - 3)(x + 4)(3x - 1)$?
First, (2x - 3)(x + 4) = 2x^2 + 5x - 12. After multiplying by (3x - 1), the x^2 terms are -2x^2 and 15x^2. Their coefficients sum to 13, so the answer is Option A.
10. Simplify: $\frac{x^3 - 8}{x^2 + 2x + 4}$
Use the sum/difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Here, $x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)$. Dividing by $(x^2 + 2x + 4)$ gives: $\frac{(x - 2)(x^2 + 2x + 4)}{x^2 + 2x + 4} = x - 2$. Therefore, the correct option is $x - 2$.
5-Minute Exam Simulation Mode
Experience the real pressure of Super TET Junior (Part 2C Mathematics). You will face 10 randomized speed-math questions with negative marking (-1 for incorrect answers) and a strict 5-minute countdown.
High-Yield Rapid Recall Formulas & Laws (Must-Remember Pointers)
Speed Math & Memory Mnemonics Vault (Exam Retention Anchors)
Polynomial Operations Rule
L-A-M-D
L - Like terms added, A - Algebraic signs distributed, M - Multiply every term, D - Degree is highest power.
Factorization Order
C-G-I
C - Common factor first, G - Grouping terms, I - Identities applied.
Negative Marking Traps & Common Blunders (-1 Penalty Avoidance)
Negative Marking Trap 1: Forgetting to distribute the negative sign when subtracting a multi-term expression.
Negative Marking Trap 2: Assuming $(a + b)^2$ equals $a^2 + b^2$ and omitting the $2ab$ cross-term.
Negative Marking Trap 3: Treating unlike terms (e.g., $x^2$ and $x$) as like terms during addition.
Negative Marking Trap 4: Misidentifying expressions with variables in the denominator as polynomials.