Polynomials — Question 1

(i) (x + 4)(x + 10)

[∵ (x + a)(x + b) = x² + (a + b)x + ab]

Putting a = 4, b = 10

= x² + (4 + 10)x + 4 x 10

= x² + 14x + 40

Hence, (x + 4)(x + 10) = x² + 14x + 40

(ii) (x + 8)(x - 10)

[∵ (x + a)(x + b) = x² + (a + b)x + ab]

Putting a = 8 , b = -10

= x² + [8 + (-10)]x + 8 x (-10)

= x² - 2x - 80

Hence, (x + 8)(x - 10) = x² - 2x - 80

(iii) (3x + 4)(3x - 5)

[∵ (x + a)(x + b) = x² + (a + b)x + ab]

Putting a = 4, b = -5, x = 3x

= (3x)² + [4 + (-5)](3x) + (4)(-5)

= 9x² + (-1)(3x) - 20

= 9x² - 3x - 20

Hence, (3x + 4) (3x - 5) = 9x² - 3x - 20

(iv) (y² + $\dfrac{3}{2}$)(y² - $\dfrac{3}{2}$)

[∵ (a + b)(a - b) = a² - b²]

Putting a = y², b = $\dfrac{3}{2}$

= (y²)² - ($\dfrac{3}{2}$)²

= y⁴ - $\dfrac{9}{4}$

Hence, (y² + $\dfrac{3}{2}$) (y² - $\dfrac{3}{2}$) = y⁴ - $\dfrac{9}{4}$

(v) (3 - 2x)(3 + 2x)

[∵ (a - b)(a + b) = a² - b²]

Putting a = 3, b = 2x

= (3)² - (2x)²

= 9 - 4x²

Hence, (3 - 2x)(3 + 2x) = 9 - 4x²