Miscellaneous examples of Algebra: part 2 for IITJEE Mains

Problem 1:

Prove that (a+b)^{5} - a^{5} - b^{5} = 5ab(a+b)(a^{2} + ab+b^{2}).

Proof I:

First carry out the following steps:

(a) Is the given expression symmetric?

(b) Is the given expression alternating?

(c) Is the given expression cyclic?

(d) Is the given expression homogeneous?

So here goes the proof:

Denote the expression by E. Clearly, the substitutions a=0, b=0 give us E=0. Hence, two of the factors are a, b. Similarly, the substitution a=-b give us E=0. So the other factor is (a+b).

The degree of  the given expression E is 5. E is symmetric w.r.t. a and b; it is homogeneous. The factors we found above are a, b, (a+b). So E=kab(a+b)F(x,y) where F(x,y) ought to be of degree 2, homogeneous and symmetric. So, let F(x, y) = Aa^{2}+Bab+Bb^{2}; thus, we have

(a+b)^{5}-a^{5}-b^{5}=kab(a+b)(Aa^{2}+Bab+Ab^{2}). Now, find k, A, and B by the undetermined coefficients.

Hence, you will get (a+b)^{5}-a^{5}-b^{5}=5ab(a+b)(a^{2}+ab+b^{2}).

Problem 2:

Find the factors of (b^{3}+c^{3})(b-c)+(c^{3}+a^{3})(c-a)+(a^{3}+b^{3})(a-b).

Solution 2:

First carry out the following steps:

(a) Is the given expression symmetric w.r.t. a and b; b and c; c and a?

(b) Is the given expression alternating w.r.t. a and b; b and c; c and a?

(c) Is the given expression cyclic w.r.t. a, b and c?

(d) Is the given expression homogeneous and if so, what is the degree of each expression?

Denote the expression by E; then, E is a function of a which vanishes when a=b, b=c, c=a, so the following are certainly factors of E: (a-b), (b-c), (c-a), and hence, E contains (a-b)(b-c)(c-a) as a factor.

Now note that E is of fourth degree and symmetric, so the remaining factor is of the form : M(a+b+c), where M is a coefficient to be determined.

So, we have now: E=M(a-b)(b-c)(c-a)(a+b+c), and you can easily find that M=1, and hence,

(b^{3}+c^{3})(b-c)+(c^{3}+a^{3})(c-a)+(a^{3}+b^{3})(a-b) = (a-b)(b-c)(c-a)(a+b+c)

More mathematical miscellany for IITJEE mathematics to follow!

Nalin Pithwa

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: