5.4 – Factoring Polynomials Completely

Key Terms

• Factor Theorem – A number a is a root of a polynomial function if and only if, when dividing the polynomial by (x – a), the remainder equals zero.
• If you get a remainder when you divide polynomials, you do NOT have a root.
• Rational Root Theorem – The rational roots of a polynomial function F(x) can be written in the form $\frac{p}{q}$, where p is a factor of the constant term of the polynomial, and q is a factor of the leading coefficient.
• Remainder Theorem – A theorem stating that when the polynomial in F(x) is divided by x – a, the remainder is F(a).
• Also, when F(x) is divided by x + a, the remainder is F(-a).

Review

• Divisibility Rules

• Roots
• Roots are the “zeros” of factors.
• Roots  are the x-intercepts of the graph.
• To find the root of a factor, set the factor equal to zero.
• Ex. $F(x)=x^2+3x+2$ has factors of $(x+1)$ and $(x+2)$
• Set the factors to equal zero and solve for x:  $x+1=0$ and $x+2=0$
• For the first factor: (x + 1), subtract 1 on both sides:  x = -1
• For the second factor (x + 2), subtract 2 on both sides: x = -2
• Lastly, test the roots using substitution.
• For F(-1)
• $F(x)=x^2+3x+2$
• $F(-1)=(-1)2+3(-1)+2$
• $F(-1)=1-3+2$
• $F(-1)=0$
• So, -1 IS a root of the polynomial.
• For F(-2)
• $F(x)=x^2+3x+2$
• $F(-2)=(-2)2+3(-2)+2$
• $F(-2)=-4-6+2$
• $F(-2)=0$
• So, -2 IS a root of the polynomial.

• Roots and Factors
• Roots have the opposite signs from their factors
• If (x + 4) is a factor, then x = -4 is a root.
• If (x – 6) is a factor, then x = 6 is a root.

• Factoring
• The constant term of a polynomial may have positive or negative factors
• The x-term of a polynomial is the sum of the factors of the constant term (which may be positive or negative).

Notes

• Factors, Roots, and Remainders
• A number a is a root of P(x) if and only if the remainder, when dividing the polynomial by x – a, equals zero.
• If a polynomial is divided by (x – a) and the remainder equals zero, then (x – a) is a factor of the polynomial.

• Roots
• If the leading coefficient is 1 (problem 2), just find all the factors of the constant term.
• If the leading coefficient is not 1 (problem 1), use the rational root theorem.
• The rational roots can be written as $\frac{p}{q}$
• p is a factor of the constant term
• q is a factor of the leading coefficient

• Coefficient of 1
• Example: $F(x)=x^2-5+14$
• Possible roots are: $\pm 1, \pm 2, \pm 7, \pm 14$

• Coefficient of a Number Other than 1
• Example:  $F(x)=3x^2-5x-2$

• Conjugates
• You know the maximum number of answers you should have.
• You know that for every radical and complex answer, you should also have its conjugate as another answer.

• Complex Roots
• For graphs, there are NO x-intercepts for complex roots.

• Factoring Polynomials
• You only need to find one root!
• You don’t have to test all the possible roots with synthetic division.
• When you find one that works (in other words, when the remainder is zero), move on to step 3
• To begin factoring, start with the roots 1, then try -1

Examples

• Example 1
• What is a root (or zero) of a polynomial function?
• Answer: A value of the variable that makes the polynomial equal to zero

• Example 2
• -6 is NOT a root because there is a remainder of 11
• (x + 6) is NOT a factor because there is a remainder of 11
• When x = -6, $2x^2+9x-7=11$
• When $2x^2+9x-7$ is divided by $x+6$, the remainder is 11

• Example 3
• 4 is a root of $F(x)=3x^2-13x+4$
• $(x-4)$ is a factor of $3x^2-13x+4$
• $(3x^2-13x+4)$ ÷ $(x-4)$ = $(3x-1)$
• $(3x^2-13x+4)$ ÷ $(3x-1)$ = $(x-4)$
• The remainder is zero