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

Answer 1

The student's question regarding rewriting a polynomial function as the product of linear factors entails performing polynomial division to divide the given function by its known factors and then factorizing the resulting quotient, if possible, into linear factors.

Step-by-step explanation:

The student asks to rewrite the polynomial function q(x)=3x^4+8x^3-13x^2-22x+24 as the product of linear factors, given that (x+3) and (x+2) are known factors of q(x).

To do this, we will carry out polynomial division to find the other factors. First, divide q(x) by (x+3) and (x+2) to obtain the quadratic factor. Once you have the quadratic factor, factorize it further to obtain two more linear factors, if possible. The end result will be q(x) rewritten as a product of four linear factors.

To demonstrate this, we would perform the following steps:

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

Divide the resulting quotient by (x+2).

Factorize the remaining quadratic equation, if factorizable.

Express q(x) as a product of the linear factors obtained.

This is a standard process when dealing with polynomial division and finding roots of polynomial equations.

Answer 2

Therefore, the complete factorization of q(x) is: q(x) = (x + 3)(x + 2)(3x + 2)(x - 5)

To rewrite this equation

We can divide polynomials since we know that (x + 3) and (x + 2) are factors of q(x):

`3x^3 - x^2 - 10x + 8x + 3 | 3x^4 + 8x^3 - 13x^2 - 22x + 24 - (3x^4 + 9x^3)`

- ----------

`-x^3 - 13x^2 - 22x + 24 + (x^3 + 3x^2)`

-----------------

`-10x^2 - 22x + 24 + (10x^2 + 30x)`

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`-52x + 24 + (52x + 156) ------------------ 180`

This division demonstrates the factorization of q(x) as::

`q(x) = (x + 3)(x + 2)(3x^2 - x - 10)`

We can also factor the remaining quadratic as well:

`3x^2 - x - 10 = (3x + 2)(x - 5)`