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Q1(iv):
Use suitable identities to find the following products:
(iv) $(y^2 + \frac{3}{2}) (y^2 – \frac{3}{2})$
Solution :
Step 1: Initial Setup & Identification of the Algebraic Identity
We are tasked with finding the product of the given binomials:
$ \left(y^2 + \frac{3}{2}\right) \left(y^2 - \frac{3}{2}\right) $
By analyzing the structural form of the expression, we observe that it consists of the product of the sum and difference of the exact same two terms. This perfectly matches the fundamental algebraic identity for the Difference of Squares:
$ (a + b)(a - b) = a^2 - b^2 $
[Theoretical Justification: The cross-terms in the expansion $(a)(a) - (a)(b) + (b)(a) - (b)(b)$ cancel out, leaving only the squared terms $a^2 - b^2$.]
Step 2: Variable Mapping and Substitution
To apply the identity rigorously, we establish a one-to-one correspondence between the variables in the identity and the terms in our specific expression.
- Let the first term $a = y^2$
- Let the second term $b = \frac{3}{2}$
Substituting these defined values into the right-hand side of the Difference of Squares identity ($a^2 - b^2$), we formulate the following equation:
$ \left(y^2\right)^2 - \left(\frac{3}{2}\right)^2 $
Step 3: Geometric Verification of the Identity (Visual Proof)
The algebraic identity $(a-b)(a+b) = a^2 - b^2$ can be geometrically proven by analyzing the area of a square of side $a$ with a smaller square of side $b$ removed. The remaining area can be rearranged into a rectangle with dimensions $(a+b)$ and $(a-b)$.
Step 4: Execution of the Algebraic Expansion & Simplification
We now simplify the expression $\left(y^2\right)^2 - \left(\frac{3}{2}\right)^2$ by applying the fundamental laws of exponents.
1. Simplifying the first term $\left(y^2\right)^2$:
According to the Power of a Power Property, $(x^m)^n = x^{m \cdot n}$. Therefore, we multiply the exponents:
$ \left(y^2\right)^2 = y^{2 \times 2} = y^4 $
2. Simplifying the second term $\left(\frac{3}{2}\right)^2$:
According to the Power of a Quotient Property, $\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$. We distribute the square to both the numerator and the denominator:
$ \left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4} $
Step 5: Final Polynomial Construction
By combining the simplified terms from Step 4 back into our expanded equation structure, we arrive at the final evaluated polynomial.
$ y^4 - \frac{9}{4} $
Final Solution: The product of the given binomials is $y^4 - \frac{9}{4}$.
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(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(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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