Extra Questions For Class 9 Maths Polynomials | Q5

Question 5: Without actual division prove that (x² - x - 2) divides 2x⁴ + x³ - 5x² - 8x - 4

Video Explanation

Explanatory Answer | Chapter 2 Extra Question 5

Approach to Solve this Polynomials Question

Let p(x) = 2x⁴ + x³ - 5x² - 8x - 4

1. Step 1: Factorize the quadratic expression (x² - x - 2) - the divisor - by splitting the middle term. Let it factorize as (x - a)(x - b)
2. Step 2: Put the zeroes 'a' and 'b' of factorizing the quadratic polynomial in p(x) and find p(a) and p(b)
3. Step 3: If p(a) and p(b) are both equal to zero, then 'a' and 'b' are zeroes of p(x)
4. Step 4: If 'a' and 'b' are zeroes of p(x), by Factor Theorem, (x - a) and (x - b) are factors of p(x)
5. Step 5: If (x - a) and (x - b) are factors of p(x), then (x - a)(x - b) will be a factor of p(x). Or, p(x) is divisible by (x² - x - 2)

Stepwise Solution

Let p(x) = 2x⁴ + x³ - 5x² - 8x - 4

Step 1: Factorize the quadratic polynomial (x² - x - 2) by splitting the middle term

x² - x - 2 = x² - 2x + x - 2
= x(x - 2) +1(x - 2)
= (x - 2)(x + 1)
Hence, zeroes of x² - x - 2 are 2 and (-1)

Step 2: Find p(2) and p(-1)

p(2) = 2(2⁴) + 2³ - 5(2²) - 8(2) - 4
= 32 + 8 - 20 - 16 - 4
0
So, 2 is a zero of p(x)

p(-1) = 2(-1)⁴ + (-1)³ - 5(-1)² - 8(-1) - 4
2 - 1 - 5 + 8 - 4
0
So, (-1) is a zero of p(x)

Step 3: Zeroes to Factors

Because 2 is a zero of p(x), (x - 2) is a factor of p(x).
Similarly, because (-1) is a zero of p(x), (x + 1) is a factor of p(x).

Step 4: The product of factors is a factor

Because (x - 2) and (x + 1) are factors of p(x), (x - 2)(x + 1) is a factor of p(x)
(x - 2)(x + 1) = x² - x - 2 is a factor of p(x).

Or, x² - x - 2 will divide p(x) without leaving a remainder.