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Q6(iii):
Write the following cubes in expanded form:
(iii) $(\frac{3}{2}x + 1)^3$
Solution :
Step 1: Identify the Applicable Algebraic Identity
To expand the given expression $(\frac{3}{2}x + 1)^3$, we utilize the standard algebraic identity for the cube of a binomial. The expansion of a binomial sum cubed is derived from multiplying the binomial by itself three times: $(a+b)(a+b)(a+b)$.
The standard identity is defined as:
$ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $
[Theoretical Justification: By the Binomial Theorem for $n=3$, the coefficients follow the sequence 1, 3, 3, 1 from Pascal's Triangle, representing the geometric decomposition of a cube into one $a^3$ volume, three $a^2b$ volumes, three $ab^2$ volumes, and one $b^3$ volume.]
Step 2: Map the Variables to the Given Expression
By comparing the given expression $(\frac{3}{2}x + 1)^3$ with the standard identity $(a + b)^3$, we establish the following variable assignments:
- $a = \frac{3}{2}x$
- $b = 1$
Step 3: Term-by-Term Expansion & Algebraic Manipulation
We will now substitute $a$ and $b$ into the identity and rigorously evaluate each of the four terms.
1. First Term ($a^3$):
$ a^3 = \left(\frac{3}{2}x\right)^3 $
$ a^3 = \frac{3^3}{2^3} \cdot x^3 = \frac{27}{8}x^3 $
[Applying the power of a product and quotient rules: $(\frac{p}{q} \cdot x)^n = \frac{p^n}{q^n} \cdot x^n$]
2. Second Term ($3a^2b$):
$ 3a^2b = 3 \cdot \left(\frac{3}{2}x\right)^2 \cdot (1) $
$ 3a^2b = 3 \cdot \left(\frac{9}{4}x^2\right) \cdot 1 $
$ 3a^2b = \frac{27}{4}x^2 $
3. Third Term ($3ab^2$):
$ 3ab^2 = 3 \cdot \left(\frac{3}{2}x\right) \cdot (1)^2 $
$ 3ab^2 = 3 \cdot \left(\frac{3}{2}x\right) \cdot 1 $
$ 3ab^2 = \frac{9}{2}x $
4. Fourth Term ($b^3$):
$ b^3 = (1)^3 = 1 $
Step 4: Synthesize the Final Polynomial
Combine the evaluated terms to form the expanded polynomial. It is standard mathematical convention to write polynomials in descending order of their degree (from the highest power of $x$ to the constant term).
$ \left(\frac{3}{2}x + 1\right)^3 = \frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1 $
Final Solution: The expanded form of the cube is $\frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1$.
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) $(–2x + 5y – 3z)^2$
- Q4(vi): Expand each of the following, using suitable identities: (vi) $(\frac{1}{4}a - \frac{1}{2}b + 1)^2$
- 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(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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