Samacheer Kalvi Class 11 Maths Solutions Chapter 2 Basic Algebra Exercise 2.7

 

Question 1. Factorize \( x^4 + 1 \)
Answer: The given expression is \( x^4 + 1 \).
We can rewrite \( x^4 + 1 \) as \( (x^2)^2 + 1^2 \).
Using the algebraic identity \( a^2 + b^2 = (a+b)^2 - 2ab \), where \( a = x^2 \) and \( b = 1 \):
\( (x^2)^2 + 1^2 = (x^2 + 1)^2 - 2(x^2)(1) \)
\( = (x^2 + 1)^2 - 2x^2 \)
Now, we can write \( 2x^2 \) as \( (\sqrt{2}x)^2 \).
So, we have \( (x^2 + 1)^2 - (\sqrt{2}x)^2 \). This is in the form \( A^2 - B^2 = (A-B)(A+B) \).
\( = (x^2 + 1 + \sqrt{2}x) (x^2 + 1 - \sqrt{2}x) \)
Finally, arrange the terms in descending order of powers of \( x \):
\( x^4 + 1 = (x^2 + \sqrt{2}x + 1) (x^2 - \sqrt{2}x + 1) \)
Factoring expressions often involves recognizing common algebraic identities.
In simple words: We take the given math expression and break it down into two simpler expressions that multiply together. This process helps simplify complex equations by using a special algebraic trick.

🎯 Exam Tip: Always look for common algebraic identities like \( a^2 + b^2 \) or \( a^2 - b^2 \) as a first step when you need to factorize expressions. This makes complex problems easier.

 

Question 2. If \( x^2 + x + 1 \) is a factor of the polynomial \( 3x^3 + 8x^2 + 8x + a \), then find the value of a.
Answer: If \( x^2 + x + 1 \) is a factor of the polynomial \( 3x^3 + 8x^2 + 8x + a \), it means that when we divide the polynomial by the factor, the remainder must be zero. We will use polynomial long division to find the value of \( a \).
We divide \( 3x^3 + 8x^2 + 8x + a \) by \( x^2 + x + 1 \):

Since the remainder must be zero for \( x^2 + x + 1 \) to be a factor:
\( a - 5 = 0 \)
\( \implies a = 5 \)
Polynomial long division is a systematic way to divide polynomials, much like how we divide numbers.
In simple words: When one math expression divides another one perfectly, there is nothing left over. We use a special division method to find the number 'a' that makes the leftover part equal to zero.

🎯 Exam Tip: When performing polynomial long division, always remember to change the signs of the terms being subtracted in each step. A common mistake is forgetting this sign change, which leads to incorrect remainders.