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Q5(ii):
Factorise:
(ii) $2x^2 + y^2 + 8z^2 – 2\sqrt{2}xy + 4\sqrt{2}yz – 8xz$
Solution :
Given Polynomial & Algebraic Foundation
We are tasked with factorising the following multivariable polynomial of degree 2:
$P(x,y,z) = 2x^2 + y^2 + 8z^2 - 2\sqrt{2}xy + 4\sqrt{2}yz - 8xz$
The structure of this expression—comprising three squared terms and three cross-product terms—directly corresponds to the algebraic identity for the square of a trinomial. [Per the standard algebraic expansion theorem]:
$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
Step 1: Extracting the Base Magnitudes
We begin by equating the pure squared terms from the given polynomial to the squared terms in the identity to find the absolute values of $a$, $b$, and $c$.
- $a^2 = 2x^2 \implies |a| = \sqrt{2}x$
- $b^2 = y^2 \implies |b| = y$
- $c^2 = 8z^2 \implies |c| = \sqrt{8}z = 2\sqrt{2}z$ [Since $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$]
Step 2: Determining the Algebraic Signs
To assign the correct positive ($+$) or negative ($-$) signs to $a$, $b$, and $c$, we must analyze the signs of the cross-product terms in the given polynomial:
| Cross-Product Term | Given Value | Sign Analysis |
|---|---|---|
| $2ab$ | $-2\sqrt{2}xy$ | Negative. [Implies $a$ and $b$ have opposite signs]. |
| $2bc$ | $+4\sqrt{2}yz$ | Positive. [Implies $b$ and $c$ have the same sign]. |
| $2ca$ | $-8xz$ | Negative. [Implies $c$ and $a$ have opposite signs]. |
From the analysis above, $b$ and $c$ share the same sign, while $a$ has the opposite sign to both. We can conventionally set $b$ and $c$ as positive, which forces $a$ to be negative. Therefore, we define our terms as:
- $a = -\sqrt{2}x$
- $b = y$
- $c = 2\sqrt{2}z$
Note: Alternatively, setting $a$ as positive forces $b$ and $c$ to be negative. Both conventions are mathematically equivalent.
Step 3: Verifying the Cross-Product Terms
Before finalizing the factorization, we must rigorously verify that our chosen terms reconstruct the original polynomial exactly.
- Check $2ab$: $2(-\sqrt{2}x)(y) = -2\sqrt{2}xy$ (Matches the given term)
- Check $2bc$: $2(y)(2\sqrt{2}z) = 4\sqrt{2}yz$ (Matches the given term)
- Check $2ca$: $2(2\sqrt{2}z)(-\sqrt{2}x) = -4(\sqrt{2} \cdot \sqrt{2})xz = -4(2)xz = -8xz$ (Matches the given term)
Step 4: Constructing the Factored Form
Since all terms perfectly align with the identity $(a + b + c)^2$, we substitute our derived values of $a$, $b$, and $c$ into the factored form.
$(a + b + c)^2 = (-\sqrt{2}x + y + 2\sqrt{2}z)^2$
To express this as a product of its factors, we write the squared binomial as the product of two identical trinomials.
Final Solution: The factorised form of the polynomial is $(-\sqrt{2}x + y + 2\sqrt{2}z)(-\sqrt{2}x + y + 2\sqrt{2}z)$.
(Note: Factoring out a $-1$ yields the equally valid alternative form: $(\sqrt{2}x - y - 2\sqrt{2}z)(\sqrt{2}x - y - 2\sqrt{2}z)$).
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) $4x^2 + 9y^2 + 16z^2 + 12xy – 24yz – 16xz$
- 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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