To calculate: The factorization of the polynomial ax2+ay+bx2+by.

Question

Unpled · Accepted Answer

Formula used:
The factors of a polynomial can be found by taking a common factor and this method is called factor by grouping,
ab+ac+bd+cd=a(b+c)+d(b+c)
=(a+d)(b+c)
Or,
ab - ac + bd - cd = a(b-c) +d(b-c)
= (a+d)(b-c)
Calculation:
Consider the polynomial,

a x^{2} + a y + b x^{2} + b y

.
This is a four terms polynomial, factorization of this polynomial can be found by factor by grouping as,

a x^{2} + a y + b x^{2} + b y = (a x^{2} + a y) + (b x^{2} + b y)

= a (x^{2} + y) + b (x^{2} + y)

As,

(x^{2} + y)

is the common factor of the polynomial factor it out as follows,

a x^{2} + a y + b x^{2} + b y = a (x^{2} + y) + b (x^{2} + y)

= (x^{2} + y) (a + b)

The factorization of the polynomial

a x^{2} + a y + b x^{2} + b y

is

(x^{2} + y) (a + b)

.
Check the result as follows,

(x^{2} + y) (a + b) = x^{2} \cdot a + x^{2} \cdot b + y \cdot a + y \cdot b

= a x^{2} + b x^{2} + a y + b y

= a x^{2} + a y + b x^{2} + b y

Thus, the factorization of the polynomial

a x^{2} + a y + b x^{2} + b y

is

(x^{2} + y) (a + b)

.

To calculate: The factorization of the polynomial ax^{2}+ay+bx^{2}+by.

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