Question

Find the value of a such that `x-2` is the factor of polynomial `x^4+ ax^3+2x^2-3x`?

Answer 1

`a=-9/4`

Explanation:

`"using the "color(blue)"factor theorem"`

`"since "(x-2)" is a factor then "f(2)=0`

`rArr2^4+a(2)^3+2(2)^2-3(2)=0`

`rArr16+8a+8-6=0`

`rArr8a=-18rArra=18/8=9/4`

Answer 2

`a = -9/4`

Explanation:

Begin by removing a factor of x:

`x^4+ ax^3+2x^2-3x = x(x^3+ax^2+2x-3)`

The reduced question is find the value of `a` such that:

`x^3+ax^2+2x-3 = (x-2)(bx^2+cx+d)`

Multiply the right side:

`x^3+ax^2+2x-3 = x(bx^2+cx+d)- 2(bx^2+cx+d)`

`x^3+ax^2+2x-3 = bx^3+cx^2+dx- 2bx^2-2cx-2d`

Combine like terms on the right:

`x^3+ax^2+2x-3 = bx^3+(c-2)x^2+(d-2c)x-2d`

We observe that `b = 1`:

`x^3+ax^2+2x-3 = x^3+(c-2)x^2+(d-2c)x-2d`

Also, `d = 3/2`

`x^3+ax^2+2x-3 = x^3+(c-2)x^2+(3/2-2c)x-3`

`2 = 3/2-2c`

`1/2=-2c`

`c = -1/4`

`x^3+ax^2+2x-3 = x^3+(-1/4-2)x^2+(3/2-2(-1/4))x-3`

`x^3+ax^2+2x-3 = x^3+(-1/4-2)x^2+2x-3`

`a = -9/4`

check:

`(x-2)(x^2-1/4x+3/2)`

`x(x^2-1/4x+3/2)-2(x^2-1/4x+3/2)`

`x^3-1/4x^2+3/2x-2x^2+1/2x-3`

`x^3+(-1/4-2)x^2+(3/2+1/2)x-3`

`x^3-9/4x^2+2x-3`

This checks.