MATH1301- Test 3
F(x)= x^1/3-2x+2
It is not a polynomial because exponent is a fraction.
If a+bi is a root of a polynomial equation with real coefficients, b≠0, then ____ is also a root of the equation.
a-bi
F(x)= -4x^4-8x^3
A. Leading Coefficient Test: The graph of f(x) falls to the left and falls to the right B. X-Intercepts: -2,0 X-Intercepts Crosses: -2,0 X-Intercepts touches and turns around at: none C. Y-Intercept: 0 D. The graph of f is neither symmetric about the y-axis nor symmetric about the origin.
All rational functions can be expressed as f(x)=p(x)/q(x), where p and q are ____ functions and q(x)≠0.
Polynomial
F(x)= -6x^2-x^3
A. Leading Coefficient Test: the graph of f(x) rises left and falls right. B. X-Intercepts: x= 0,-6 X-Intercept crosses: -6 X-Intercept touches and turns around at: 0 C. Y-intercept: 0 D. Neither
If a polynomial equation is of degree n, then counting multiple roots separately, the equation has ____ roots.
n
f(x)=x^4+x^8+2 1.Determine whether the function is even, odd, or neither. 2. Determine whether the graph of the function is symmetric with respect to the y-axis, the origin, or neither.
1. Even 2. y-axis
The perimeter of a rectangle is 90 feet. If the length of the rectangle is represented by x, its width can be expressed as ____. The area of the rectangle, A(x), expressed in the form A(x)=ax^2+bx+c, is A(x)=____.
45-x , -x^2+45x
The degree of the polynomial function f(x)=-2x^3(x-1)(x+5) is ____. The leading coefficient is ____.
5, -2
f(x)=1-(x-3)^2
Axis of Symmetry: x=3 Domain: (-infinity, +infinity) Range: (-infinity, 1]
f(x)=2x-x^2+8
Axis of symmetry: x=1 D: (-infinity,+infinity) R: (-infinity,9]
f(x)=(x-4)^2-9
Axis of symmetry: x=4 Domain: (-infinity, +infinity) Range: [-9, +infinity)
After performing polynomial long division, the answer may be checked by multiplying the ____ by the ____, and then adding the ____. This should give the ____
Divisor, Quotient, Remainder, Dividend
The vertex is (-4, -8) and the parabola opens up.
Domain: (-infinity,+infinity) Range: [-8,+infinity)
The remainder theorem states that if a polynomial F(x) is divided by (x-a), then the remainder R=____.
F(a)
A polynomial function with four sign changes must have four positive real zeros.
FALSE
The graph of f(x)=(x+7)^2+6 has its vertex at (7,6).
FALSE
The graph of f(x)=-x^2 ____.
Falls to the left and falls to the right.
The graph of f(x)=x^3 ____ to the left and ____ to the right.
Falls, rises
G(x)=6x^6-piex^3+1/7x
It is a polynomial. The degree of the polynomial is 6.
The graph of a rational function has a slant asymptote if the degree of the numerator is ____ the degree of the denominator.
One more than
Write the point-slope form of the line's equation satisfying the given conditions. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope=5, passing through (3,2)
Point-slope form of equation: y-2=5(x-3) Slope-intercept of equation: y=5x-13
A polynomial function with seven sign changes can have one, three, five, or seven positive real zeros.
TRUE
If the degree of the numerator of a rational function equals the degree of the denominator, then the ratio of the leading coefficient gives rise to the horizontal asymptote.
TRUE
Odd-degree polynomial functions have graphs with opposite behavior at each end.
TRUE
The graph of f(x)= -2(x+3)^2-5 opens down. State whether the statement is true or false.
TRUE
The y-coordinate of the vertex of f(x)=2x^2-20x+254 is f(5).
TRUE
The y-coordinate of the vertex of f(x)=3x^2-60x+400 is f(10).
TRUE
F(x)= -3x^4+7x^2-x+2
The graph of f(x) falls to the left and falls to the right.
If r is a zero of even multiplicity, then the graph touches the x-axis and ____ at r. If r is a zero of odd multiplicity, then the graph ____ the x-axis at r.
Turns around, crosses
F(x)= 2(x-8)(x+5)^3
Zero(s): 8, -5 Multiplicities of zero: Largest is 1, Smallest is 3 The behavior of the function at each zero: The graph crosses the x-axis at all zeros
Consider the polynomial function with integer coefficients f(x)=anxn+an−1xn−1+•••+a1x+a0, an≠0. The Rational Zero Theorem states that if p/q is a rational zero of f (where p/q is reduced to lowest terms), then p is a factor of ____ and q is a factor of ____.
a0, an
The quadratic function f(x)=a(x−h)^2+k, a≠0, is in ____ form. The graph of f is called a ____ whose vertex is the point ____. The graph opens upward if ____ and opens downward if ____.
standard, parabola, (h,k), a>0, a<0
f(x)=-3(x+2)^2+9
vertex: (-2,9)
f(x)=4(x-5)^2-2
vertex: (5,-2)
The Factor Theorem states that if f is a polynomial function and f(c)=0, then ____ is a factor of f(x).
x-c