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