# Factorise the algebraic expression ${z^2} - z - 2$.

Answer

Verified

282k+ views

**Hint:**For factoring the given expression we have to find the roots or follow factorization process.

Method 1: FACTORIZATION: In this process we spit the z term into the sum of any two appropriate numbers whose product results in the constant value of the expression. From here we can simplify the rest of the expression.

Method2: GETTING ROOTS: In this method, we find the roots of the expression. We already know that for roots a, b the expression is $(x - a)(x - b)$ and its elaborate form is ${x^2} - (a + b)x + ab$.

Here in this question we take roots be a, b. As from the equation above we know the sum of roots and product of roots . We can get a-b from the formula

\[{(a - b)^2} = {(a + b)^2} - 4ab\]

Now from a + b and a – b we get the values of a and b and substitute them in $(x - a)(x - b)$.

**Complete step-by-step solution:**

METHOD 1: -z can be written as the sum of -2z and z.

${z^2} - z - 2 \\

= {z^2} - 2z + z - 2 \\

= z(z - 2) + (z - 2) \\

= (z + 1)(z - 2) $

Hence, ${z^2} - z - 2 = (z + 1)(z - 2)$.

METHOD 2 :

Let us consider the roots of expression ${z^2} - z - 2$ be a, b.

By comparison with ${x^2} - (a + b)x + ab$,

$a + b = 1 \\

ab = - 2 $

Then,

${(a - b)^2} = {(1)^2} - 4( - 2) \\

\Rightarrow {(a - b)^2} = 1 + 8 \\

\Rightarrow {(a - b)^2} = 9 \\

\Rightarrow a - b = 3 \\

\Rightarrow a = 3 + b $

Substitute the arrived value ‘a’ in a + b , we get

$a + b = 1 \\

\Rightarrow 3 + b + b = 1 \\

\Rightarrow 3 + 2b = 1 \\

\Rightarrow 2b = - 2 \\

\Rightarrow b = - 1 $

and

$ a = 3 + b \\

\Rightarrow a = 3 + ( - 1) \\

\Rightarrow a = 2 $

The expression

\[{z^2} - z - 2 \\

= (z - a)(z - b) \\

= (z - 2)(z - ( - 1)) \\

= (z - 2)(z + 1) \]

**Note:**The graph of expression ${z^2} - z - 2$ is

From the graph the roots are ( -1 , 0 ) , ( 2 , 0 ) . From the graph the minimum of the expression ${z^2} - z - 2$ is ( 0.5 , -2.25 ) . Graphical methods are another way to find the roots. The points where the curve intersects the X-axis are roots.

Recently Updated Pages

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

What is 1 divided by 0 class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the past tense of read class 10 english CBSE

What is pollution? How many types of pollution? Define it

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

How many crores make 10 million class 7 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE