Question

The polynomial function q(x)= 3x^4 + 8x^3 - 13x^2 - 22x + 24 has known factors (x + 3) and (x + 2). Rewrite q(x) as the product of linear factors

q(x) = (x + 3)(x + 2)(____)(____)

Answer 1

Simplify the following:

Simplify the following:3 x^4 + 8 x^3 - 13 x^2 - 22 x + 24

Simplify the following:3 x^4 + 8 x^3 - 13 x^2 - 22 x + 24The possible rational roots of 3 x^4 + 8 x^3 - 13 x^2 - 22 x + 24 are x = ± 1/3, x = ± 2/3, x = ± 4/3, x = ± 8/3, x = ± 1, x = ± 2, x = ± 3, x = ± 4, x = ± 6, x = ± 8, x = ± 12, x = ± 24.

Simplify the following:3 x^4 + 8 x^3 - 13 x^2 - 22 x + 24The possible rational roots of 3 x^4 + 8 x^3 - 13 x^2 - 22 x + 24 are x = ± 1/3, x = ± 2/3, x = ± 4/3, x = ± 8/3, x = ± 1, x = ± 2, x = ± 3, x = ± 4, x = ± 6, x = ± 8, x = ± 12, x = ± 24. Of these, x = 4/3, x = 1, x = -2 and x = -3 are roots. This gives 3 x - 4, x - 1, x + 2 and x + 3 as all factors:

Simplify the following:3 x^4 + 8 x^3 - 13 x^2 - 22 x + 24The possible rational roots of 3 x^4 + 8 x^3 - 13 x^2 - 22 x + 24 are x = ± 1/3, x = ± 2/3, x = ± 4/3, x = ± 8/3, x = ± 1, x = ± 2, x = ± 3, x = ± 4, x = ± 6, x = ± 8, x = ± 12, x = ± 24. Of these, x = 4/3, x = 1, x = -2 and x = -3 are roots. This gives 3 x - 4, x - 1, x + 2 and x + 3 as all factors:Answer: (3 x - 4) (x - 1) (x + 2) (x + 3)

Answer 2

`q(x) = (x + 3)(x + 2)(3x-4)(x-1)`

Explanation:

Given that the polynomial function q(x)= 3x⁴ + 8x³ - 13x² - 22x + 24 has known linear factors (x + 3) and (x + 2), we can find the other linear factors by first dividing q(x) by one of the factors, then dividing the quotient by the other factor. We can then factor the resulting quadratic.

Divide q(x) by (x + 3):

`\large \begin{array}{r}3x^3-x^2-10x+8\phantom{)}\\x+3{\overline{\smash{\big)}\,3x^4 + 8x^3 - 13x^2 - 22x + 24\phantom{)}}\\{-~\phantom{(}\underline{(3x^4+9x^3)\phantom{-bwwwwwwwww)}}\\-x^3-13x^2-22x+24\phantom{)}\\-~\phantom{()}\underline{(-x^3-3x^2)\phantom{wwwwwww}}\\-10x^2-22x+24\phantom{)}\\-~\phantom{()}\underline{(-10x^2-30x)\phantom{www}}\\8x+24\phantom{)}\\-~\phantom{()}\underline{(8x+24)}\\0\phantom{)}\end{array}`

Therefore, q(x) = (x + 3)(3x³ - x² - 10x + 8).

Now divide the cubic factor (3x³ - x² - 10x + 8) by the other known linear factor (x + 2):

`\large \begin{array}{r}3x^2-7x+4\phantom{)}\\x+2{\overline{\smash{\big)}\,3x^3-x^2-10x+8\phantom{)}}\\{-~\phantom{(}\underline{(3x^3+6x^2)\phantom{-www.b)}}\\-7x^2-10x+8\phantom{)}\\-~\phantom{()}\underline{(-7x^2-14x+8)}\\4x+8\phantom{)}\\-~\phantom{()}\underline{(4x+8)}\\0\phantom{)}\end{array}`

Therefore, q(x) = (x + 3)(x + 2)(3x² - 7x + 4).

Now, factor the quadratic factor (3x² - 7x + 4) to find the other two linear factors:

`\begin{aligned}3x^2-7x+4&=3x^2-3x-4x+4\\&=3x(x-1)-4(x-1)\\&=(3x-4)(x-1)\end{aligned}`

Therefore, the polynomial function q(x) as the product of linear factors is:

`q(x) = (x + 3)(x + 2)(3x-4)(x-1)`