Multiply x² + 4y² + z² + 2xy + xz - 2yz by (- z + x - 2y).

Solution:

Given, the expression is x² + 4y² + z² + 2xy + xz - 2yz

We have to multiply the expression by (-z + x - 2y)

(-z + x - 2y) (x² + 4y² + z² + 2xy + xz - 2yz) = -z(x² + 4y² + z² + 2xy + xz - 2yz) + x(x² + 4y² + z² + 2xy + xz - 2yz) - 2y(x² + 4y² + z² + 2xy + xz - 2yz)

Now, -z(x² + 4y² + z² + 2xy + xz - 2yz) = x²(-z) + 4y²(-z) + z²(-z) + 2xy(-z) + xz(-z) - 2yz(-z)

= -zx² - 4y²z - z³ - 2xyz - xz² + 2yz²

x(x² + 4y² + z² + 2xy + xz - 2yz) = x²(x) + 4y²(x) + z²(x) + 2xy(x) + xz(x) - 2yz(x)

= x³ + 4xy² + xz² + 2x²y + x²z - 2xyz

-2y(x² + 4y² + z² + 2xy + xz - 2yz) = x²(-2y) + 4y²(-2y) + z²(-2y) + 2xy(-2y) + xz(-2y) - 2yz(-2y)

-2x²y - 8y³ - 2yz² - 4xy² - 2xyz + 4y²z

Now, (-z + x - 2y) (x² + 4y² + z² + 2xy + xz - 2yz) = -zx² - 4y²z - z³ - 2xyz - xz² + 2yz² + x³ + 4xy² + xz² + 2x²y + x²z - 2xyz - 2x²y - 8y³ - 2yz² - 4xy² - 2xyz + 4y²z

- 2xyz - 2xyz - 2xyz

By grouping,

= - z³ + x³ - 8y³ + 2x²y - 2x²y - 4xy² + 4xy² + 4y²z - 4y²z - 2yz² + 2yz² + x²z - x²z - xz² + xz² - 2xyz - 2xyz - 2xyz

= - z³ + x³ - 8y³ - 2xyz - 2xyz - 2xyz

= - z³ + x³ - 8y³ - 6xyz

Therefore, (-z + x - 2y) (x² + 4y² + z² + 2xy + xz - 2yz) = x³ - 8y³ - z³ - 6xyz

✦ Try This: Multiply 3x² + 2y² + 4z² + 2xy + 3xz - 6yz by (- 2z + 2x - 2y).

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 2

NCERT Exemplar Class 9 Maths Exercise 2.4 Problem 6

Multiply x² + 4y² + z² + 2xy + xz - 2yz by (- z + x - 2y)

Summary:

On multiplying (-z + x - 2y) (x² + 4y² + z² + 2xy + xz - 2yz) we get x³ - 8y³ - z³ - 6xyz by grouping the terms

☛ Related Questions:

• If a, b, c are all non-zero and a + b + c = 0, prove that a²/bc + b²/ca + c²/ab = 3
• If a + b + c = 5 and ab + bc + ca = 10, then prove that a³ + b³ + c³ - 3abc = - 25
• Prove that (a + b + c)³ - a³ - b³ - c³ = 3(a + b ) (b + c) (c + a)