Question

Prove that factorisation is the reverse of multiplication.

Explain clearly with the help of example.

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Answer 1

TO PROVE:

The factorisation is the reverse of multiplication.

PROOF:

Let us take (x + 10)²

(A + B)² = A² + 2AB + B²

(x + 10)² = x² + 2x(10) + 10²

(x + 10)² = x² + 20x + 100

This is expanding or otherwise called multliplication. We can clearly see the expanded form of (x + 10)²

(x + 10)² cannot be futher shortened. But x² + 20x + 100 can be further shortened which in other words is called as factorization.

By splitting the middle term

x² + 10x+ 10x + 100

x(x + 10) + 10(x + 10)

(x + 10)(x + 10)

(x + 10)²

We again got the same term as (x + 10)² which on expanding gave x² + 20x + 100 and on again factorizing gave (x + 10)².

Here we can clearly see that multiplication is the antonym of factorization or vice-versa.

HENCE PROVED THAT THE FACTORIZATION IS THE OPPOSITE OF MULTIPLICATION.

Answer 2

$\huge\sf\blue{Given}$

✭ Factorisation is an opposite of Multiplication

$\rule{110}1$

$\huge\sf\gray{To\;Prove}$

◈ The above statement holds True

$\rule{110}1$

$\huge\sf\purple{Steps}$

So here first what is factorisation.. it is the shortening of some big thing and as an opposite in multiplication we expand something which in its short form.For Example

$\sf Take (x+5)^2$

➝ $\sf x^2+2(x)(5)+5^2$

$\sf \small\quad\bigg\lgroup \because (x+y)^2 = x^2+2xy+y^2\bigg\rgroup$

➝ $\sf x^2+10x+25$

So here we shall bring it back to its old form by factorising

➳ $\sf x^2+10x+25$

$\sf 25 \qquad | \qquad 10\\ \underline{ \qquad \qquad \qquad \quad} \\ \sf 5 \times 5 \: \: \: | \: \: 5 + 5\\ \underline{\qquad\qquad\qquad\quad}$

➳ $\sf x^2+5x+5x+25$

➳ $\sf x(x+5)+5(x+5)$

➳ $\sf (x+5)(x+5)$

➳ $\sf (x+5)^2$

Actually,just like we call that addition is opposite of subtraction, Similarly factorisation is just a reverse of multiplication

$\sf Hence \ Proved !!$

$\rule{170}3$