Question

Factorize
i)a^3 – 2a^2– 4a + 8
ii) (ax + by)^2 + (bx – ay)^2
​

Answer 1

$\textsf{\large{\underline{Solution}:}}$

Q1)

Given:

$\rm=a^{3}-2a^{2}-4a+8$

Take a² as common from first two terms, we get:

$\rm=a^{2}(a-2)-4a+8$

Take -4 as common from last two terms, we get:

$\rm=a^{2}(a-2)-4(a-2)$

Take (a - 2) as common:

$\rm=(a^{2}-4)(a-2)$

Using identity a² - b² = (a + b)(a - b), we get:

$\rm=(a+2)(a-2)(a-2)$

$\rm=(a+2)(a-2)^{2}$

Which is our required answer.

————————————————————————————————

Q2)

Given:

$\rm=(ax+by)^{2}+(bx-ay)^{2}$

Expanding the terms, we get:

$\rm=(ax)^{2}+(by)^{2}+2axby+(bx)^{2}+(ay)^{2}-2axby$

$\rm=(ax)^{2}+(by)^{2}+(bx)^{2}+(ay)^{2}$

On rearranging the terms, we get:

$\rm=(ax)^{2}+(bx)^{2}+(ay)^{2}+(by)^{2}$

Taking x² as common from first two terms, we get:

$\rm=x^{2}(a^{2}+b^{2})+(ay)^{2}+(by)^{2}$

Taking u² as common from last two terms, we get:

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

Taking (a² + b²) as common, we get:

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

Which is our required answer.

$\textsf{\large{\underline{Answer}:}}$

1. (a + 2)(a - 2)²
2. (x² + y²)(a² + b²)

Answer 2

Answer:

$\huge{\bf{\underline{\underline{ \colorbox{black}{\color{white}{Answer}}}}}}$

Step-by-step explanation:

$[tex]\textsf{\large{\underline{Solution}:}}$

Q1)

Given:

$\rm=a^{3}-2a^{2}-4a+8$

Take a² as common from first two terms, we get:

$\rm=a^{2}(a-2)-4a+8$

Take -4 as common from last two terms, we get:

$\rm=a^{2}(a-2)-4(a-2)$

Take (a - 2) as common:

$\rm=(a^{2}-4)(a-2)$

Using identity a² - b² = (a + b)(a - b), we get:

$\rm=(a+2)(a-2)(a-2)$

$\rm=(a+2)(a-2)^{2}$

Which is our required answer.

————————————————————————————————

Q2)

Given:

$\rm=(ax+by)^{2}+(bx-ay)^{2}$

Expanding the terms, we get:

$\rm=(ax)^{2}+(by)^{2}+2axby+(bx)^{2}+(ay)^{2}-2axby$

$\rm=(ax)^{2}+(by)^{2}+(bx)^{2}+(ay)^{2}$

On rearranging the terms, we get:

$\rm=(ax)^{2}+(bx)^{2}+(ay)^{2}+(by)^{2}$

Taking x² as common from first two terms, we get:

$\rm=x^{2}(a^{2}+b^{2})+(ay)^{2}+(by)^{2}$

Taking u² as common from last two terms, we get:

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

Taking (a² + b²) as common, we get:

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

Which is our required answer.

$\textsf{\large{\underline{Answer}:}}$

(a + 2)(a - 2)²

(x² + y²)(a² + b²)[/tex]