Question

Find the values of `a` and `b` so that the polynomial `(x^4+ax^3-7x^2-8x+b)` is exactly divisible by`(x+2)` as well as`(x+3)`?

Answer 1

`a = 2`

`b = 12`

Explanation:

Multiply the two factors `(x+2)(x+3)`:

`(x+2)(x+3) = x^2+ 3x + 2x +6`

`(x+2)(x+3) = x^2+ 5x +6`

This multiplies with another trinomial to produce the given quartic polynomial:

`(x^2+ 5x +6)(x^2+cx+d) = x^4+ax^3-7x^2-8x+b`

Multiply each term of the first factor by the second factor:

`x^2(x^2+cx+d)+ 5x(x^2+cx+d)+6(x^2+cx+d) = x^4+ax^3-7x^2-8x+b`

Use the distributive property:

`x^4+cx^3+dx^2+ 5x^3+5cx^2+5dx+6x^2+6cx+6d = x^2+ax^3-7x^2-8x+b`

Use the `x^3` terms to write equation [1]:

`a = c+5" [1]"`

Use the `x^2` terms to write equation [2]:

`d+5c + 6 = -7" [2]"`

Use the x terms to write equation [3]:

`5d+6c = -8" [3]"`

Use the constant terms to write equation [4]:

`b = 6d" [4]"`

Use equation 2 to express `d` in terms of `c`:

`d+5c + 6 = -7" [2]"`

`d = -5c-13" [2.1]"`

Substitute equation [2.1] into equation [3]:

`5(-5c-13)+6c = -8`

`-25c-65+6c = -8`

`-19c = 57`

`c = -3`

Use equation [2.1] to find the value of d:

`d = -5(-3)-13`

`d = 2`

Use equation [1] to find the value of a:

`a = -3+5`

`a = 2`

Use equation [4] to find the value of b:

`b = 6(2)`

`b = 12`