Higher Degree Equations, Solve & Graph; hanlonmath

End Behaviors In even functions, as x → + infinity y = ax² + ... y = ax⁴ + ...

y → + infinity

End Behaviors In even functions, as x → - infinity

y → + infinity

End Behaviors In odd functions, as x→ + infinity, y = ax + ... y = ax³ + ... y = ax⁵ + ...

y → + infinity

End Behaviors In odd functions, as x→ - infinity

y → - infinity

Procedure for Synthetic Substitution

1. Write the coefficients of the polynomials; fill in place holders 2. Write the value you are substituting in a box 3. Draw a line 4. Bring down the leading coefficient 5. Multiply the leading coefficient by the value being evaluated 6. Add that result to the next coefficient CONTUNUE that process

In the polynomial; P(x) = 4x³ - 5x + 3, write the coefficients you would use to do synthetic substitution

4, 0, -5, 3 Since there is no quadratic term, 0 is the placeholder for that term

How can you determine the degree of a factored polynomial?

Add the exponents for each factor

In P(x) = (x - 1)(x - 5)²(x +7)³, find the degree

Degree 6; 1 + 2 + 3

When graphing, how can it be determined if the graph crosses the x-axis for a particular zero

EVEN multiplicity - does not cross - bounces ODD multiplicity - crosses

In the polynomial, P(x) = x³ + 5x² - 7x + 6 = 0 List all the possible rational roots

Factorsof 6 are ±1, 2, 3, 6; -p Factors of 1 are ±1; - q Possible solutions; p/q are ±1, 2, 3, 6

Procedure to determine if a (x - c) is a factor

First, change the sign of "c", then follow the same procedure for Synthetic Substitution. If the remainder is zero, then (x - c) is a factor

Remainder Theorem

For every polynomial P(x) of positive degree n over the set of complex numbers, and for every complex number r, there exists a polynomial Q(x) of degree (n-1) such that: P(x) = (x-r)Q(x) + P(r) When a polynomial, P(x) is divided by (x - c), the remainder is P(c). When you substitute "c" into P, the remainder is the y-coordicate of the ordered pair. (c, P(c))

Rational Root Theorem

If P(x) = 0 is a polynomial equation with integral coefficients of degree n in which a₀ is the coefficient of xⁿ, and a₄ is the constant term, then for any rational root p/q, p is a factor of the constant and q is a factor of the leading coefficient. If there is a rational root, it has to come from the factors of the constant over the factors of the leading coefficient.

Factor Theorem

Over the set of complex numbers, (x - c) is a factor of a polynomial P(x) if and only if P(c) = 0 The factor theorem is a special case of the Remainder Theorem. It also tells us where the graph crosses the x-axis; roots or zeros

In P(x) = (x - 1)(x - 5)²(x +7)³, what is the multiplicity of the factor (x - 5)? of (x + 7)

The multiplicity of (x - 5) is 2; won't cross the x-axis The multiplicity of (x + 7) is 3, will cross x-axis

End Behaviors What happens to end behaviors when the leading coefficient is negative

The y values are opposite of end values when a is positive