Equations and inequalities
Solve 5x + 2 > 7x - 4.
x < 3
The quadratic x2+7x+12 can be factored as (x+a)(x+b), where a and b are positive integers.
What is the value of a×b?
12
For the polynomial P(x)=3x5−2x4+7x2−9:
a) What is the degree of the polynomial?
b) What is the leading coefficient?
c) What is the constant term?
Degree = 5, Leading coefficient = 3, Constant term = -9
Given the polynomial function f(x)=2x3−3x2+5x−7, find f(2).
f(2)=7
8−2/3
1/4
Solve |x - 5|= 7.
X = -2, X = 12
One root of the quadratic equation 2x2+kx−15=0 is x=3.
Find the value of k and the other root.
k=−1, other root x=−5/2
Given P(x)=2x3−5x2+3x−7, find P(2).
What does this value represent in terms of polynomial division?
P(2)=−5; this is the remainder when dividing by x−2
Find all zeros of the polynomial function f(x)=x2−5x+6.
x=2 and x=3
Find the domain f(x)=(x−5)1/2
[5,∞)
Solve |2x + 12| = 4x. Check for extraneous solutions.
6=x
A quadratic function f(x)=ax2+bx+c has its vertex at (2,3) and passes through the point (4,11).
Find the value of a+b+c.
5
The polynomial P(x)=x3+kx2−4x+12 has a factor x+3.
Find the value of k.
k=1/3
Find a polynomial function of degree 3 that has zeros at x=−2, x=1, and x=4.
f(x)=x3−3x2−6x+8.
Find the domain g(x)=(3x+2)1/3
(−∞,∞)
Write an equation of the line that passes through (-2, 3) and is perpendicular to the line y = -4x + 1.
y= 1/4x+7/2
The quadratic equation x2+6x+k=0 has two distinct real roots.
Find the range of possible values for k.
k<9
The polynomial P(x)=x3−7x+6 has a factor x−2.
Find all three factors of P(x). Hint: Use synthetic division.
(x−2)(x+3)(x−1)
A polynomial function has zeros at x=−1, x=2, and x=3. The function passes through the point (1,−12). Find the y-intercept of the function.
−18
Simplify completely. Express your answer with positive exponents only.
(8x6/27y−3)2/3
4x4y2/9
A linear function f(x)=ax+b satisfies the following conditions:
f(1)=5
f(f(1))=17
Find the value of f(2).
f(2)=8
The quadratic function f(x)=x2−5x+k has two distinct real roots, and both roots are positive.Find the range of possible values for k.
0<k<6.25
Factor the polynomial completely:
P(x)=x4−5x3+5x2+5x−6
(x−1)(x+1)(x−2)(x−3)
Consider the polynomial function: f(x)=−2(x+3)2(x−1)(x−4)3
a) What is the degree of the function?
b) Describe the end behavior (what happens as x→∞ and x→−∞)
Degree = 6, As x→∞, f(x)→−∞, As x→−∞, f(x)→−∞
Solve for x: (x+7)1/2=x−5, Check for extraneous solutions.
x=9