Question

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

Answer 1

Solution :-

Let us assume that, (x - 1) is a factor of given polynomial. so, using remainder theorem we get,

→ f(x) = 2x³ - 7x² - 10x +24

→ f(1) = 2(1)³ - 7(1)² - 10(1) + 24

→ f(1) = 2 - 7 - 10 + 24

→ f(1) = 5

checking for (x + 2) ,

→ f(-2) = 2(-2)³ - 7(-2)² - 10(-2) + 24

→ f(-2) = -16 - 28 + 20 + 24

→ f(-2) = 0

then, (x + 2) is a factor of given polynomial .

dividing now we get,

x + 2 ) 2x³ - 7x² - 10x + 24 ( 2x² - 11x + 12

2x³ + 4x²

-11x² - 10x

-11x² - 22x

12x + 24

12x + 24

0

with the help of long division method now, dividing 2x³ - 7x² - 10x + 24 by x + 2 using synthetic division method we get, (x +2 so , we will take (-2) as divisor.) from above divisible avoid all variables first, and then avoid all the partial products. Then, we get,

-2) 2 , (-7) , (-10) , 24 (

2 4

-11 - 22

12 24

0

finally , we get,

-2) 2 , (-7) , (-10) , 24 (

2 , (-11) , 12 , 0

therefore, we get,

→ Quotient = 2x² - 11x + 12

→ Remainder = 0 . { Last digit is remainder . }

hence,

→ 2x³ - 7x² - 10x +24

→ (x + 2)(2x² - 11x + 12)

→ (x + 2)[2x² - 8x - 3x + 12]

→ (x + 2)[2x(x - 4) - 3(x - 4)]

→ (x + 2)(x - 4)(2x - 3) (Ans.)

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Answer 2

SOLUTION

TO DETERMINE

To factorise the polynomial 2x³ - 7x² - 10x + 24 using synthetic division method

EVALUATION

Here the given polynomial is

$\sf{p(x) = 2 {x}^{3} - 7 {x}^{2} - 10x + 24 }$

Now we first find the first factor as below

$\sf{p(1) = 2. {1}^{3} - 7 .{1}^{2} - 10.1 + 24 = 9 \ne \: 0}$

$\sf{p( - 1) = 2. {( - 1)}^{3} - 7 .{( - 1)}^{2} - 10.( - 1) + 24 = 25 \ne \: 0}$

$\sf{p(2) = 2. {2}^{3} - 7 .{2}^{2} - 10.2 + 24 = - 8 \ne \: 0}$

$\sf{p( - 2) = 2. {( - 2)}^{3} - 7 .{( - 2)}^{2} - 10.( - 2) + 24 = 0}$

So - 2 is one of the zeroes of p(x)

Thus ( x + 2 ) is one of factors of p(x)

We now apply Synthetic Division method as below :

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{ - 2}}}&{{\sf{\:\:2\: \: \: - 7 \: \: \: - 10 \: \: \: \: \: \: \: 24 \:\:}}}\\ \\ {\underline{\sf{}}}& \underline{\sf{\:\:\: \: 0 \: \: \: - 4 \: \: \: \: \: \: \: 22 \: \: \: - 24 \:\:}} \\ {\underline{\sf{ 4}}}&{{\sf{ \: \: \: \: 2\: \: \: - 11 \: \: \: \: \: \: 12\: \: \: \: \: \: \: \boxed{0} \:\:}}}\\ \\ {\underline{\sf{}}}& \underline{\sf{ 0 \: \: \: \: \ \: \: \: \: \: 8 \: \: \: - 12 \: \: \: \: \: \: \: \:\:}}\\{\sf{}}&{\sf{ \: \: 2\: \: \: \: \: \: \: - 3 \: \: \: \: \: \: \boxed{0} \: \: \: \: \: \: \: \: \: \: \:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

In 2nd step we see that ( x - 4 ) is another factor

In 3rd step we see that ( 2x - 3 ) is another factor

So the all three factors are ( x + 2 ) , ( x - 4 ) , ( 2x - 3 )

Thus on factorisation we have

$\sf{2 {x}^{3} - 7 {x}^{2} - 10x + 24 = (x + 2)(x - 4)(2x - 3) }$

FINAL ANSWER

$\boxed{ \: \: \sf{2 {x}^{3} - 7 {x}^{2} - 10x + 24 = (x + 2)(x - 4)(2x - 3) } \: \: }$

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