Question

show by long division method x - 3 is a factor of 2x⁴+3x³-26x²-5x+6

Answer 1

as the answer comes in zero therefore it's a factor of px

Answer 2

Yes, x - 3 is a factor of 2x⁴+3x³-26x²-5x+6.

Given:

• $2 {x}^{4} + 3 {x}^{3} - 26 {x}^{2} - 5x + 6$
• (x-3)

To find:

• Show that x-3 is a factor of $2 {x}^{4} + 3 {x}^{3} - 26 {x}^{2} - 5x + 6$ by long division.

Solution:

Concept to be used:

If remainder is zero then g(x) is a factor of p(x).

Step 1:

Let

$p(x) = 2 {x}^{4} + 3 {x}^{3} - 26 {x}^{2} - 5x + 6$

and

$g(x) = x - 3$

Step 2:

Perform long division.

$x - 3 \: ) \: 2 {x}^{4} + 3 {x}^{3} - 26 {x}^{2} - 5x + 6 \: (2 {x}^{3} + 9 {x}^{2} + x - 2 \\ 2 {x}^{4} - 6 {x}^{3} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ ( - ) \: \: \: ( + ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ - - - - - - - \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ 9 {x}^{3} - 26 {x}^{2} \\ 9 {x}^{3} - 27 {x}^{2} \\ ( - ) \: \: ( + ) \: \: \: \\ - - - - - - \\ {x}^{2} - 5x \\ {x}^{2} - 3x \\ ( - ) \: \: ( + ) \\ - - - - - - \\ - 2x + 6 \\ - 2x + 6 \\ ( + ) \: \: ( - ) \\ - - - - - \\ 0 \\ - - - - -$

Here,

Remainder of division is zero.

Thus,

g(x) is a factor of p(x).

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