Algebraic Identities | Polynomials | Assessment | English | Grade 9

(x - y)³ = ________________

• x³- y³ + 3x²y - 3xy²

• x³- y³- 3x²y + 3xy²

• x³- y³ + 3x²y + 3xy²

• x³+ y³- 3x²y - 3xy²

Answer explanation

(x - y)³ = x³- y³- 3x²y + 3xy²

Expand (4x - 6y - 10z)²

• 4x² + 6y² + 10z² - 48xy + 120yz - 80xz

• 16x² + 36y² + 48xy + 120y

• 16x² + 36y² + 100z² - 48xy + 120yz

• 16x² + 36y² + 100z² - 48xy + 120yz - 80xz

Answer explanation

Here, we will use the identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2xz So, (4x - 6y - 10z)² = [4x + (-6y) + (-10z)]² = (4x)² + (-6y)² + (-10z)² + 2(4x)(-6y) + 2(-6y)(-10z) + 2(4x)(-10z) = 16x² + 36y² + 100z² - 48xy + 120yz - 80xz

Factorize: x(x + y)³ - 3x²y (x + y)

• x(x - y)(x²+ y²+ xy)

• x(x - y)(x²+ y²-xy)

• x(x - y)(x²- y²-xy)

• x(x + y)(x²+ y²-xy)

Answer explanation

x(x + y)³ - 3x²y (x + y) = x(x + y) {(x + y)² - 3xy } = x(x + y) {(x² + y²+2xy - 3xy)} = x(x + y)(x²+ y²-xy)

Factorize: x² - y² + 2yz - z²

• (x + y -z) (x - y + z)

• (x - y + z) (x - y + z)

• (x + y + z) (x - y + z)

• (x + y - z) (x + y - z)

Answer explanation

x² - y² + 2yz - z² By rearranging the terms we will get, = x² - (y² + z² - 2yz) Now we can use the identity, (a - b)² = (a² - 2ab + b²) = x² - (y - z)² We know that, a² - b² = (a + b) (a - b) Here, a = x and b = y - z ∴ (x + y - z) (x - y + z) Hence we can say that option 1 the correct option.

109² - 106² = ?

• 645

• 654

• 612

• 600

Answer explanation

Here, we will use the identity x² - y² = (x + y) (x - y) 109² - 106² = (109 + 106) (109 - 106) = 215(3) = 645

If 81a² - b = (9a + 1/2) (9a - 1/2), then find the value of b.

• 1/3

• 1/2

• 1/4

• 1

Answer explanation

81a² - b = (9a + 1/2) (9a - 1/2) We can write 81a² - b as (9a)² - (√b)² = (9a + √b) (9a - √b) (9a + √b) (9a - √b) = (9a + 1/2) (9a - 1/2) On comparing, √b = 1/2 Hence, b = 1/4

Find the length and breadth of a rectangle if it's area is represented by polynomial 6x² - 29x + 30

• (2x + 3) and (3x + 10)

• (2x - 3) and (3x - 10)

• (2x + 3) and (3x - 10)

• (2x - 3) and (3x + 10)

Answer explanation

Area of a rectangle = length × breadth We are given, area of rectangle = 6x² - 29x + 30 By splitting the middle term we will get 6x² - 9x - 20x + 30 = 3x(2x - 3) - 10 (2x - 3) = (3x - 10) (2x - 3) Hence, (3x - 10) and (2x - 3) are possible length and breadth of a rectangle whose area is 6x² - 29x + 30.