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Free Homework Math Help: Cubic and Higher Degree Polynomials

 

Cubic and Higher Degree Polynomials

Going from expanded form to factored form uses factoring while distribution is used to go from factored to expanded form

Just like with quadratic equations, higher degree polynomials have two forms: expanded and factored. The following example uses distribution to expand a cubic polynomial in factored form. A cubic polynomial is a polynomial of degree 3.

             (x+1)(x+2)(x+3) three factors

Distribute the terms of the first factor into the second factor.

Binomial factors are multiplied together one factor at a time

A new factor is created that is the expansion of the first two factors. Repeat the process above of distributing the terms of the first factor into the second factor.

The terms of the quadratic factor are distributed to the terms of the linear factor

Now the polynomial is in expanded form.

Another example shows two cubic equations being multiplied together by distribution.

Two cubic polynomials are multiplied together by distributing the terms of one factor into the second

Factoring a Polynomial

The process of factoring a polynomial is more complicated. The simplest method to use is factoring by grouping. To factor:

             x3 + 3x2 + 4x + 12
             x2(x+3) + 4(x+3)
             (x2+4)(x+3)

Substitution can also be used:

             x4 + 2x2 + 1      y = x2
             y2 + 2y + 1 = (y+1)2 = (x2+1)2

Another method uses the Rational Zero Theorem:

Rational Zero Theorem

       Given the polynomial:
P(x) = anxn+an-1xn-1+...+a2x2+a1x+a0,
if the coefficients a1 ... an are integers and p/q is a rational zero of P(x) then:
1. p is a factor of a0
2. q is a factor of an

The constant and the leading coefficient are factored and the p/q ratios are written out. At this point, each result is substituted back into the original polynomial. If the substitution yields a zero, then p/q is a root of the polynomial.