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