How do you factor #x^3-4x^2-11x+30#?

Question

Avery Alexander · Accepted Answer

#=>p(x)=(x-2)(x-5)(x+3)#

Here,

#p(x)=x^3-4x^2-11x+30#

For #x=2#,

#p(2)=2^3-4(2)^2-11(2)+30=8-16-22+30=0#

#=>(x-2)#, is a factor.

#:.p(x)=x^3-2x^2-2x^2+4x-15x+30#

#=>p(x)=x^2color(green)((x-2))-2xcolor(green)((x-2))-15color(green)((x-2))#

#=>p(x)=(x-2)(x^2color(red)(-2x)-15)#

#=>p(x)=(x-2)[x^2color(red)(-5x+3x)-15]#

#=>p(x)=(x-2)[xcolor(blue)((x-5))+3color(blue)((x-5))]#

#=>p(x)=(x-2)(x-5)(x+3)#

Eli Assouline · Answer

undefined To factor the expression $x^3 - 4x^2 - 11x + 30$, you can use various factoring methods. One common approach is to look for rational roots using the Rational Root Theorem and synthetic division.

First, list all possible rational roots using the Rational Root Theorem. The possible roots are the factors of the constant term (30) divided by the factors of the leading coefficient (1). So, the possible rational roots are ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30.

Next, use synthetic division or substitution to find the roots. Once you find a root, use synthetic division to divide the polynomial by the root. Repeat this process until you have factored the polynomial completely.

Alternatively, you can use grouping or other factoring techniques if you notice any patterns or common factors among the terms.

Once you find one root, you can use polynomial long division or synthetic division to divide the polynomial by the corresponding linear factor, which will give you a quadratic expression. Then, you can further factor the quadratic expression if possible.

Continue this process until you have factored the polynomial completely into linear factors.

The factored form of $x^3 - 4x^2 - 11x + 30$ will be a product of linear factors, such as $(x - a)(x - b)(x - c)$, where $a$, $b$, and $c$ are the roots of the polynomial.