Question

factorize the following polynomials using synthetic division method 2x3-7x2-10x+24​

Answer 1

we have to factorize the polynomial 2x³ - 7x² - 10x + 24 using synthetic division method.

solution : first write it in coefficient form,

2 , -7 , -10 , 24

checking for x = 1

1 | 2, -7, -10, 24

1 -6 -16

................................

2 -6 -16 8 [ not zero ] so x = 1 is not zero of given polynomial.

let's check for x = 2

2 | 2, -7, -10, 24

4. -6 -8

...............................

2 -3 -4 16 [ not zero ]

again, check for x = -2

-2 |2, -7, -10, 24

-4 22 -24

................................

2 -11 12 0 [ zero so x = -2 is zero of given polynomial and (2x² - 11x + 12) is another factor of given polynomial ]

so, factorisation is (x + 2)(2x² - 11x + 12)

you can also apply, synthetic division method for 2x² - 11x + 12

take x = 4, as f(x) = 2x² - 11x + 12 = 0 at x = 4

4 |2 , - 11, 12

8. -12

.......................

2 -3 0

so 2x² - 11x + 12 = (x + 4)(2x - 3)

Therefore factorisation of 2x³ -7x² - 10x + 24 = (x + 2)(x + 4)(2x - 3)

Answer 2

Step-by-step explanation:

First we need to do trial and error method

put x=-2

= 2(-2)^3-7(-2)^2-10(-2)+24

= 2(-8)-7(4)+20+24

= -16-28+20+24

= 0

So, x+2 is a factor

Now, use synthetic division method

coefficients of the polynomial are

2,-7,-10,24

x=-2 | 2. -7. -10. 24

+ | 0. -4. 22. -24. ( -2× the result)

2. -11. 12. 0 -- Remainder

x^2. x. constant

Quotient = 2x^2-11x+12

Use Factorisation

2x^2-11x+12

= 2x^2-8x-3x+12

= 2x(x-4)-3(x-4)

= (x-4)(2x-3)

Factors are (x+2)(x-4)(2x-3)