Chapter 3 Polynomial Fucntion

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Chapter 3 Polynomial Fucntion

Factoring Sum or Difference of Cubes

Ex: 27x^3+125=(3x+5)(9x^2-15x+25)

a^3-b^3 = (a-b)(a^2+ab+b^2)

a^3+b^3 = (a+b)(a^2-ab+b^2)

Factoring Polynomial

Lastly, divide f(x) by (x-a)

Try each of them until f(a)=0 If f(a)=0, then (x-a) is a factor of f(x)

Possible values for "a" are the factors of the constant term (term with no variable) in f(x) Ex: factors of 9 are +-1, +-2, +-3

To factor f(x) we need to find (x-a) where f(a)=0

The Remainder Theorem

Ex: divide f(x)=x^2-4 by x-2 f(2)=(2)^2-2=0 Therefore, (x-2) is a factor of f(x) meaning remainder is 0

When a polynomial f(x) is divided by (x-a), the remainder is equal to f(a) If the remainder is equal to zero, then (x-a) is a factor of f(x)

Dividing Polynomial

Division Statement: Divisor * Quotient + Remainder

Synthetic Division

7. The last number(10) on the bottom row is the remainder and the rest are the co-efficient of the quotient.

6. Multiply the last number below the line by k and repeat step 3-5

5. Multiply the number below the line by k. Put the product above the line in the second column and then add it with the co-efficient above it 3*4=12 -2+12=10

4. Drop the first co-efficient

3. Copy the co-efficient inside the chart and place k or k/a to the left

1./2. Repeat step 1 and 2 from Long Division

Synthetic division can only be used when the divisor is linear Ex: (x-k)

Long Division

6. Repeat steps 3-5

5. Subtract the dividend with the product

4. Multiply the last term of quotient by the divisor, write the result below the dividend 3x*(2x+5)

3. Divide the first term of the dividend by the first term of the divisor 6x^2/2x=3x

2. Fill any missing terms from the general form of polynomial with zeroes

1. Write both the divisor and dividend in descending powers of the variables

End Behavior

For EVEN degree function

If leading co-efficient is negative, graph extends from 3rd quadrant to 4th quadrant

x → +-∞, y → -∞

If leading co-efficient is positive, graph extends from 2nd quadrant to 1st quadrant

x → +-∞, y → ∞

For ODD degree function

If leading co-efficient is negative, graph extends from 2nd quadrant to 4th quadrant

x → -∞, y → +∞ x → +∞, y → -∞

If leading co-efficient is positive, graph extends from 3rd quadrant to 1st quadrant.

x → -∞, y → -∞ x → +∞, y → +∞

Characteristic of Function

Leading Co-Efficient

is the co-efficient of the highest power

Symmetry

Most polynomial functions are neither

Odd function has symmetry over both the x and y-axis

Even function has symmetry over the y-axis

Order in Factored form

Order 3 is cubed(cubic shape)

Order 2 is quadratic(parabolic)

Order 1 is linear(curve crosses through x-axis)

Order of an x-intercept is its exponent Ex: (x-3)^2 meaning x=3 has the order of 2

In factored form, we can find x-intercept

Roots

Function with even degree may have no roots

Function with odd degree must have at least 1 root because it has opposite end bahavior

Arms and Turning Points

Number of arms of a function is determined by its degree(most of the time)

An arm is a continuous lines in between 2 turning points

Number of turning points of a function is at most degree-1

Turning Points is the where the function change vertical direction

Degree

5th degree is quintic

4th degree is quartic

3rd degree is cubic

2nd degree is quadratic

1st degree is linear

is the exponent of the highest power