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Q3(i):
Factorise the following using appropriate identities:
(i) $9x^2 + 6xy + y^2$
Solution :
Initial Setup & Expression Analysis
We are tasked with factorising the following algebraic polynomial:
$9x^2 + 6xy + y^2$
To factorise this expression efficiently, we must analyze its structural properties. The polynomial consists of three terms (a trinomial). Our primary objective is to determine if it conforms to the structure of a perfect square trinomial, which can be factorised using standard algebraic identities.
Step 1: Decomposing the Terms into Perfect Squares
We begin by examining the first and third terms to see if they can be expressed as perfect squares [Per the fundamental properties of exponents].
- First Term: The term $9x^2$ consists of a constant $9$ and a variable $x^2$. Since $3^2 = 9$, we can rewrite this term as the square of a single monomial:
$9x^2 = (3x)^2$ - Third Term: The term $y^2$ is already a perfect square and can be explicitly written as:
$y^2 = (y)^2$
Step 2: Verifying the Middle Term
For a trinomial to be classified as a perfect square trinomial, the middle term must be exactly twice the product of the bases of the squared terms identified in Step 1.
Let us define our base variables based on the standard identity structure:
- Let $a = 3x$
- Let $b = y$
Now, we calculate the theoretical middle term, $2ab$:
$2ab = 2 \cdot (3x) \cdot (y)$
$2ab = 6xy$
This calculated product perfectly matches the middle term of our given polynomial ($6xy$). Therefore, the expression is confirmed to be a perfect square trinomial.
Step 3: Applying the Standard Algebraic Identity
Having verified the structure, we apply the First Algebraic Identity for the square of a binomial [By the binomial expansion theorem]:
$a^2 + 2ab + b^2 = (a + b)^2$
Substituting our specific values ($a = 3x$ and $b = y$) into the identity:
$(3x)^2 + 2(3x)(y) + (y)^2 = (3x + y)^2$
Expanding the squared binomial into its linear factors yields:
$(3x + y)^2 = (3x + y)(3x + y)$
Geometric Proof & Visualization
The algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$ can be rigorously proven using area models in Euclidean geometry. Below is the geometric representation of our specific polynomial, where the total area of a square with side length $(3x + y)$ is equal to the sum of the areas of its four internal rectangular regions.
As demonstrated by the geometric model, the total area is the sum of the individual regions: $9x^2 + 3xy + 3xy + y^2 = 9x^2 + 6xy + y^2$, which perfectly corresponds to a square of side $(3x + y)$.
Final Solution: The completely factorised form of the polynomial $9x^2 + 6xy + y^2$ is $(3x + y)(3x + y)$, which is most concisely written as $(3x + y)^2$.
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(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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