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Q4(vi):
Expand each of the following, using suitable identities:
(vi) $(\frac{1}{4}a - \frac{1}{2}b + 1)^2$
Solution :
Given Expression & Algebraic Identity
We are tasked with expanding the following trinomial squared:
$ \left( \frac{1}{4}a - \frac{1}{2}b + 1 \right)^2 $
To expand this expression systematically, we utilize the standard algebraic identity for the square of a trinomial [Derived from the distributive property of multiplication over addition]:
$ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx $
Step 1: Variable Mapping
By comparing our given expression $\left( \frac{1}{4}a - \frac{1}{2}b + 1 \right)^2$ with the standard identity $(x + y + z)^2$, we can establish a direct mapping of the terms. It is critical to include the negative sign in our mapping to maintain algebraic integrity.
- $ x = \frac{1}{4}a $
- $ y = -\frac{1}{2}b $
- $ z = 1 $
Step 2: Substitution into the Identity
Substituting the mapped variables into the expanded form of the identity, we get:
$ \left( \frac{1}{4}a \right)^2 + \left( -\frac{1}{2}b \right)^2 + (1)^2 + 2\left( \frac{1}{4}a \right)\left( -\frac{1}{2}b \right) + 2\left( -\frac{1}{2}b \right)(1) + 2(1)\left( \frac{1}{4}a \right) $
Step 3: Term-by-Term Simplification
We will now apply the exponent rules [specifically $(uv)^n = u^n v^n$] and perform scalar multiplication for each distinct term.
| Component | Operation | Simplified Result |
|---|---|---|
| $x^2$ | $\left( \frac{1}{4}a \right)^2$ | $\frac{1}{16}a^2$ |
| $y^2$ | $\left( -\frac{1}{2}b \right)^2$ | $\frac{1}{4}b^2$ [Note: The square of a negative is positive] |
| $z^2$ | $(1)^2$ | $1$ |
| $2xy$ | $2 \cdot \left( \frac{1}{4}a \right) \cdot \left( -\frac{1}{2}b \right)$ | $-\frac{1}{4}ab$ |
| $2yz$ | $2 \cdot \left( -\frac{1}{2}b \right) \cdot (1)$ | $-b$ |
| $2zx$ | $2 \cdot (1) \cdot \left( \frac{1}{4}a \right)$ | $\frac{1}{2}a$ |
Step 4: Final Assembly
Combining all the simplified terms from Step 3 yields the fully expanded polynomial. We write the terms in descending order of degree where applicable, though standard expansion order is perfectly rigorous:
$ \frac{1}{16}a^2 + \frac{1}{4}b^2 + 1 - \frac{1}{4}ab - b + \frac{1}{2}a $
Final Solution: The expanded form of the given polynomial is $ \frac{1}{16}a^2 + \frac{1}{4}b^2 + 1 - \frac{1}{4}ab - b + \frac{1}{2}a $
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.4
- Q1(i): Use suitable identities to find the following products: (i) $(x + 4) (x + 10)$
- Q1(ii): Use suitable identities to find the following products: (ii) $(x + 8) (x – 10)$
- Q1(iii): Use suitable identities to find the following products: (iii) $(3x + 4) (3x – 5)$
- Q1(iv): Use suitable identities to find the following products: (iv) $(y^2 + \frac{3}{2}) (y^2 – \frac{3}{2})$
- Q1(v): Use suitable identities to find the following products: (v) $(3 – 2x) (3 + 2x)$
- Q10(i): Factorise each of the following: (i) $27y^3 + 125z^3$ [Hint : See Question 9.]
- Q10(ii): Factorise each of the following: (ii) $64m^3 – 343n^3$ [Hint : See Question 9.]
- Q11: Factorise : $27x^3 + y^3 + z^3 – 9xyz$
- Q12: Verify that $x^3 + y^3 + z^3 – 3xyz = \frac{1}{2}(x + y + z)[(x – y)^2 + (y – z)^2 + (z – x)^2]$
- Q13: If $x + y + z = 0$, show that $x^3 + y^3 + z^3 = 3xyz$.
- Q14(i): Without actually calculating the cubes, find the value of each of the following: (i) $(–12)^3 + (7)^3 + (5)^3$
- Q14(ii): Without actually calculating the cubes, find the value of each of the following: (ii) $(28)^3 + (–15)^3 + (–13)^3$
- Q15(i): Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (i) Area : $25a^2 – 35a + 12$
- Q15(ii): Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (ii) Area : $35y^2 + 13y –12$
- Q16(i): What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (i) Volume : $3x^2 – 12x$
- Q16(ii): What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (ii) Volume : $12ky^2 + 8ky – 20k$
- Q2(i): Evaluate the following products without multiplying directly: (i) $103 \times 107$
- Q2(ii): Evaluate the following products without multiplying directly: (ii) $95 \times 96$
- Q2(iii): Evaluate the following products without multiplying directly: (iii) $104 \times 96$
- Q3(i): Factorise the following using appropriate identities: (i) $9x^2 + 6xy + y^2$
- Q3(ii): Factorise the following using appropriate identities: (ii) $4y^2 – 4y + 1$
- Q3(iii): Factorise the following using appropriate identities: (iii) $x^2 – \frac{y^2}{100}$
- Q4(i): Expand each of the following, using suitable identities: (i) $(x + 2y + 4z)^2$
- Q4(ii): Expand each of the following, using suitable identities: (ii) $(2x – y + z)^2$
- Q4(iii): Expand each of the following, using suitable identities: (iii) $(–2x + 3y + 2z)^2$
- Q4(iv): Expand each of the following, using suitable identities: (iv) $(3a – 7b – c)^2$
- Q4(v): Expand each of the following, using suitable identities: (v) $(–2x + 5y – 3z)^2$
- Q5(i): Factorise: (i) $4x^2 + 9y^2 + 16z^2 + 12xy – 24yz – 16xz$
- Q5(ii): Factorise: (ii) $2x^2 + y^2 + 8z^2 – 2\sqrt{2}xy + 4\sqrt{2}yz – 8xz$
- Q6(i): Write the following cubes in expanded form: (i) $(2x + 1)^3$
- Q6(ii): Write the following cubes in expanded form: (ii) $(2a – 3b)^3$
- Q6(iii): Write the following cubes in expanded form: (iii) $(\frac{3}{2}x + 1)^3$
- Q6(iv): Write the following cubes in expanded form: (iv) $(x - \frac{2}{3}y)^3$
- Q7(i): Evaluate the following using suitable identities: (i) $(99)^3$
- Q7(ii): Evaluate the following using suitable identities: (ii) $(102)^3$
- Q7(iii): Evaluate the following using suitable identities: (iii) $(998)^3$
- Q8(i): Factorise each of the following: (i) $8a^3 + b^3 + 12a^2b + 6ab^2$
- Q8(ii): Factorise each of the following: (ii) $8a^3 – b^3 – 12a^2b + 6ab^2$
- Q8(iii): Factorise each of the following: (iii) $27 – 125a^3 – 135a + 225a^2$
- Q8(iv): Factorise each of the following: (iv) $64a^3 – 27b^3 – 144a^2b + 108ab^2$
- Q8(v): Factorise each of the following: (v) $27p^3 – \frac{1}{216} – \frac{9}{2}p^2 + \frac{1}{4}p$
- Q9(i): Verify : (i) $x^3 + y^3 = (x + y) (x^2 – xy + y^2)$
- Q9(ii): Verify : (ii) $x^3 – y^3 = (x – y) (x^2 + xy + y^2)$
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
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