NCERT Solutions Class 8 Maths Algebraic Expressions and Identities Exercise 9.1

Key Terminology in Algebraic Expressions

Term

The parts of an algebraic expression that are added or subtracted. For example, in 3x + 5y, '3x' and '5y' are terms.

Factor

The variables or numbers that are multiplied together to build a term. For example, the term 5xy has factors 5, x, and y.

Coefficient

The numerical factor in an algebraic term. For example, in the term -7x^2y, the coefficient is -7.

Like Terms

Terms that contain the exact same variables raised to the exact same powers. Only like terms can be added or subtracted.

The Secret of Like and Unlike Terms

Algebraic terms are like families. Terms that have the exact same variable parts (including their powers) are called like terms. For example, $5xy$ and $-3xy$ are like terms because they both share the variable part $xy$. However, $5xy$ and $5x^2y$ are unlike terms because the power of $x$ is different. This distinction is crucial because in algebra, we can only add or subtract like terms! It is just like counting real-world items: 3 apples plus 2 apples equals 5 apples, but 3 apples plus 2 bananas cannot be combined into a single fruit type—they remain 3 apples and 2 bananas. When adding expressions, group the like terms together and simply add their numerical coefficients while keeping the variable part unchanged.

Step-by-Step: Adding and Subtracting Algebraic Expressions

1. Identify and Write Down the Expressions — Write down the given expressions clearly. Pay close attention to the signs (+ or -) belonging to each individual term.
2. Align Like Terms — Use either the Horizontal Method (grouping like terms in brackets) or the Column Method (writing one expression below the other so that like terms fall in the same column).
3. Apply Sign Rules (For Subtraction Only) — If you are subtracting, change the sign of every term in the expression that is being subtracted. Positive terms become negative, and negative terms become positive.
4. Simplify and Calculate — Perform addition or subtraction on the numerical coefficients of the grouped like terms to find the final simplified expression.

The Negative Sign Trap in Subtraction

The most common mistake students make in algebraic expressions and identities ex 9 1 class 8 ncert happens during subtraction. When asked to subtract Expression A from Expression B, you must write it as: $\text{Expression B} - (\text{Expression A})$. The negative sign outside the parenthesis applies to every single term inside Expression A. For example, subtracting $(3x - 4y)$ from $(5x + 2y)$ gives: $(5x + 2y) - (3x - 4y) = 5x + 2y - 3x + 4y = 2x + 6y$. Notice how the sign of $-4y$ flipped to $+4y$. Always use parentheses to protect your signs!

Practice Questions with Solutions

• Q: Identify the terms and their coefficients for the algebraic expression: $7x^2y^2 - 5x^2y^2z^2 + 3z^2$. A: Step 1: Break down the expression into its individual terms. The terms are $7x^2y^2$, $-5x^2y^2z^2$, and $3z^2$. Step 2: Identify the numerical factor (coefficient) for each term. - For $7x^2y^2$, the coefficient is $7$. - For $-5x^2y^2z^2$, the coefficient is $-5$ (make sure to include the negative sign). - For $3z^2$, the coefficient is $3$. Final answer: Terms: $7x^2y^2$ (coef: 7), $-5x^2y^2z^2$ (coef: -5), $3z^2$ (coef: 3).
• Q: Classify the following expressions as monomials, binomials, or trinomials: (i) $y^2$, (ii) $5 - 3t$, (iii) $x + y + z^2$, (iv) $ab + bc + cd + da$. A: Step 1: Recall the classification based on the number of non-zero terms: - 1 term: Monomial - 2 terms: Binomial - 3 terms: Trinomial - More than 3 terms: Polynomial Step 2: (i) $y^2$ has 1 term. Hence, it is a Monomial. Step 3: (ii) $5 - 3t$ has 2 terms. Hence, it is a Binomial. Step 4: (iii) $x + y + z^2$ has 3 terms. Hence, it is a Trinomial. Step 5: (iv) $ab + bc + cd + da$ has 4 terms. This does not fit into the first three specific types; it is classified generally as a Polynomial. Final answer: (i) Monomial, (ii) Binomial, (iii) Trinomial, (iv) Polynomial.
• Q: Add the following algebraic expressions: $ab - bc$, $bc - ca$, and $ca - ab$. A: Step 1: Write down the expressions as a sum: $(ab - bc) + (bc - ca) + (ca - ab)$. Step 2: Group the like terms together: $(ab - ab) + (-bc + bc) + (-ca + ca)$. Step 3: Simplify each grouped pair: $(1-1)ab + (-1+1)bc + (-1+1)ca = 0 + 0 + 0 = 0$. Final answer: The sum is $0$.
• Q: Subtract $4a - 7ab + 3b + 12$ from $12a - 9ab + 5b - 3$. A: Step 1: Set up the subtraction order: $(12a - 9ab + 5b - 3) - (4a - 7ab + 3b + 12)$. Step 2: Distribute the negative sign to all terms inside the second bracket: $12a - 9ab + 5b - 3 - 4a + 7ab - 3b - 12$. Step 3: Group the like terms together: $(12a - 4a) + (-9ab + 7ab) + (5b - 3b) + (-3 - 12)$. Step 4: Solve the arithmetic for each group: $8a - 2ab + 2b - 15$. Final answer: $8a - 2ab + 2b - 15$.

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