Quick Crisp Review

• Zeros of a polynomial function are where the x-intercepts or solutions when you set the equation equal to zero.

• Synthetic and long division

You will be able to used the remainder theorem and rational roots theorem.

If a polynomial f(x) is divided by (x – a) then the remainder is equal to f(a).

If the remainder is 0 then (x – a) is a factor of the polynomial.

Let f(x) = x4 – 6x3 + 8x2 + 5x + 13, evaluate f(4).

Use Direct Substitution.

Use Synthetic Division (The answer is the remainder)

Is (x – 4) a factor?

Is (x – 3) a factor ofx3 + 5x2 – 12x – 36?Factor Theorem

(x – a) is a factor of f(x), if f(a) = 0

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The rational roots theorem states that every rational zero is equal to p/q, where p is a factor of the constant and q is a factor of the leading coefficient.

What are the possible roots of f(x) = 5x3 - 6x2 – 9x + 2?Factors of p are

Factors of q are

5

2,5

1,2,1

Determine the zeros off(x) = 5x3 - 6x2 –9x + 2?

The possible rational roots were

5

2,5

1,2,1

When a polynomial is divided by x – a, where a>0, if all terms of the quotient and the remainder are positive then a is an upper bound.

When a polynomial is divided by x – a, where a<0, if all terms of the quotient and remainder alternate in sign then a is a lower bound.

Consider the function p(x) = 2x3 – 11x2 – 10x + 55

Is 7 an upper bound?

Is -3 a lower bound?

Exit

• Explain how the theorems allow you to determine the zeros without graphing?

• Page 193 #56

ACT: If x2 + 6x + 8 = 4 + 10x, then x equals which of the following?

A) -2 B) -1 C) 0 D) 1 E) 2