Math Chapter 2

What is the Fundamental Theorem of Algebra?

If f(x) is a polynomial of degree n, where n>0, then f has at least one zero in the complex number system

What is the Linear Factorization Theorem?

If f(x) is a polynomial of degree n, where n>0, then f has precisely n linear factors

When 0<a<1, the graph of y=af(x) is a..... of the graph of y=f(x)?

When 0<a<1, the graph of y=af(x) is a vertical shrink of the graph of y=f(x)

i⁵=?

i

What is the equation for the Rational Zero Test?

possible rational zeros=factors of the constant term/factors of the leading coefficient

What is the equation of the Quadratic Formula?

x= -b±√b²-4ac/2a

If a is a positive number, then the principle square root of the negative number -a is defined as...?

√-a=i√a

What are second degree polynomial functions called?

Quadratic functions

Multiply: (2-i)(4+3i)

2(4+3i)-i(4+3i) 8+6i-4i-3i² 8+3+6i-4i 11+2i

What are the graphs of polynomial functions of degree 1?

Lines

The remainder r, obtained in the synthetic division of a polynomial f(x) by x-k, provides what 3 pieces of information?

1. The remainder r gives the value of f at x=k. That is, r=f(k) 2. If r=0, then (x-k) is a factor of f(x) 3. If r=0, then (k,0) is an x-intercept of the graph of f

How are polynomial functions classified?

Classified by degree

Write the polynomial f(x)=x⁴-x²-20 in completely factored form

x⁴-x²-20=(x+√5)(x-√5)(x-2i)(x+2i)

Write the polynomial f(x)=x⁴-x²-20 as the product of linear factors and quadratic factors that are irreducible over the reals

x⁴-x²-20=(x+√5)(x-√5)(x²+4)

What is the definition of a Polynomial Function?

(The ₀ really represents n) Let n be a nonnegative integer and let a₀, a₀−₁, a₂, a₁... be real numbers with a₀≠0. The function given by a₀x⁰+a₀-₁x⁰-¹...+a₂x²+a₁x+a₀ is called a polynomial function of x with degree n.

What is the equation for the difference of two complex numbers a+bi and c+di?

(a+bi)-(c+di)=(a-c)-(b-d)i

i²=?

-1

What are the characteristics of a polynomial function?

-It is continuous (no breaks, holes, or gaps) -Has only smooth, rounded turns (no sharp turns)

i³=?

-i

What are the coordinates of the vertex of f in variable form?

(-b/2a, f(-b/2a))

What is the equation for the sum of two complex numbers a+bi and c+di?

(a+bi)+(c+di)=(a+c)+(b+d)i

i⁴=?

1

Let f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is divided by x-c, using synthetic division. 1. If c>0...? 2. If c<0..?

1. If c>0 and each number in the last row is either positive or zero, then c is an upper bound for the real zeros of f 2. If c<0 and the numbers in the last row are alternately positive and negative, then c is a lower bound for the real zeros of f

In the standard form a+bi, what is the number bi called?

The imaginary part of the complex number a+bi

Write the standard form of the equation of the parabola whose vertex is (1,2) and that passes through the point (3,-6)

f(x)=-2(x-1)²+2

Use Synthetic Division to divide x⁴-10x²-2x+4 by x+3

x⁴-10x²-2x+4/x+3=x³-3x²-x+1+1/x+3

For a polynomial function, a factor of (x-a)^k, k>1, yields a repeated zero x=a of multiplicity k. If k is even, then the graph...?

If k is even, then the graph touches the x-axis (but does not cross the x-axis) at x=a

When n is even and the leading coefficient is negative, then the graph...?

The graph falls to the left and right

When n is odd and the leading coefficient is positive, then the graph...?

The graph falls to the left and rises to the right

What is the graph of a constant function?

The graph of a constant function is a horizontal line.

Describe the graph of f(x)=-x²+6x-8 and identify any x-intercepts

The graph of f is a parabola that opens downward with vertex (3,1). The x-intercepts are (2,0) and (4,0)

Describe the graph of f(x)=2x²+8x+7 and identify the vertex

The graph of f is a parabola that opens upward with a vertex (-2, -1). Corresponds to a left shift of 2 units and a downward shift of 1 unit relative to the graph of y=2x²

When n is odd and the leading coefficient is negative, then the graph...?

The graph rises to the left and falls to the right

When n is even and the leading coefficient is positive, then the graph...?

The graph rises to the left and right

When a>1, the graph of y=af(x) is a..... of the graph of y=f(x)?

When a>1, the graph of y=af(x) is a vertical stretch of the graph of y=f(x)

For the general quadratic form f(x)=ax²+bx+c, when the leading coefficient a is negative....?

When the leading coefficient a is negative, the parabola opens downward and the vertex is the highest point

For the general quadratic form f(x)=ax²+bx+c, when the leading coefficient a is positive....?

When the leading coefficient a is positive, the parabola opens upward and the vertex is the lowest point

What can you test with the Factor Theorem?

You can test whether a polynomial has (x-k) as a factor by evaluating the polynomial at x=k. If the result is 0, then (x-k) is a factor

What is the complex conjugate of a+bi?

a-bi

Graph of Parent Cubic Function f(x)=x³ Domain? Range? Intercept? Increasing on? Even or odd? Symmetry?

Domain: (-inf, inf) Range: (-inf, inf) Intercept: (0,0) Increasing on: (-inf, inf) Even or odd: Odd Symmetry: Origin symmetry

What is the definition of a Quadratic Function?

Let a, b, and c be real numbers with a≠0. The function given by f(x)=ax²+bx+c is called a quadratic function

If a<0, then f has a maximum at...? The maximum value is...?

Maximum at x=-b/2a, the minimum value is f(-b/2a)

If a>0, then f has a minimum at...? The minimum value is...?

Minimum at x=-b/2a, the minimum value is f(-b/2a)

How do you write the quotient of a+bi and c+di in standard form?

Multiply the numerator and denominator by the complex conjugate of the denominator

When |a| is large...?

The parabola opens more narrowly

When |a| is small...?

The parabola opens more widely

What is a Constant Function?

The polynomial function f(x)=a, a≠0 has degree 0 and is called a constant function.

What is a Linear Function?

The polynomial function f(x)=mx+b, m≠0 has degree 1 and is called a linear function

What is the equation for the Standard Form of a quadratic function?

The quadratic function given by f(x)=a(x-h)²+k, a≠0 is in Standard Form

Example of a function that is irreducible over the reals?

The quadratic x²+1=(x-i)(x+i) is irreducible over the reals (and therefore over the rationals)

Example of a function that is irreducible over the rationals?

The quadratic x²-2=(x-√2)(x+√2) is irreducible over the rationals, but reducible over the reals.

What is the equation for the Division Algorithm?

f(x)/d(x)=q(x)+r(x)/d(x)

i=?

√-1

It can be shown that for a polynomial function f of degree n, what two statements are true?

1. The function f has at most n real zeros 2. The graph of f has at most n-1 relative minima or maxima

What is Decartes's Rule of Signs?

1. The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer 2. The number of negative real zeros of f is either equal to the number of variations in sigh of f(-x) or less than that number by an even integer

What are the two hints for finding zeros?

1. When the terms of f(x) have a common monomial factor, it should be factored out before applying the tests. 2. When you are able to find all but two zeros of f, you can always use the Quadratic Formula on the remaining quadratic factor.

Before you apply the Division Algorithm, what steps do you need to follow?

1. Write the dividend and divisor in descending powers of the variable 2. Insert placeholders with 0 coefficients for missing powers of the variable

When f is a polynomial function and a is a real number, what four statements are equivalent?

1. x=a is a zero of function f 2. x=a is a solution of the polynomial equation f(x)=0 3. (x-a) is a factor of the polynomial f(x) 4. (a,0) is an x-intercept of the graph of f

Use Long Division to divide -2+3x-5x²+4x³+2x⁴ by x²+2x-3

2x⁴+4x³-5x²+3x-2/x²+2x-3=2x²+1+x+1/x²+2x-3

Use long division to divide the polynomial 6x³-19x²+16x-4 by x-2 and use the results to factor the polynomial completely. What are the x-intercepts?

6x³-19x²+16x-4=(x-2)(2x-1)(3x-2) he 3 x-intercepts occur at x=2, x=1/2 and x=2/3

Use Long Division to divide 8x³-1 by 2x-1

8x³-1/2x-1=4x²+2x+1, x≠1/2

What is the graph of a linear function?

A line whose slope is m and whose y-intercept is (0, b)

What is the Factor Theorem?

A polynomial f(x) has a factor (x-k) if and only if f(k)=0

Use the Remainder Theorem to evaluate f(x)=3x³+8x²+5x-7 when x=-2

Because the remainder is r=-9, you can conclude that f(-2)=-9. This means that (-2, -9) is a point on the graph of f.

Show that (x-2) and (x+3) are factors of f(x)=2x⁴+7x³-4x²-27x-18. Then find the remaining factors.

Complete factorization: f(x)=(x-2)(x+3)(2x+3)(x+1)

How many zeros does the fifth degree polynomial function f(x)=x⁵+9x³ have? What are the zeros?

Exactly five zeros: x=0, x=0, x=3i, and x=-3i

How many zeros does the first degree polynomial f(x)=x-2 have? What is the zero?

Exactly one zero: x=2

How many zeros does the third degree polynomial function f(x)=x³+4x have? What are the zeros?

Exactly three zeros: x=0, x=2i, and x=-2i

How many zeros does the second degree polynomial function f(x)=x²-6x+9 have? What are the zeros?

Exactly two zeros: x=3 and x=3

Use long division to divide x²+3x+5 by x+1. What is the fractional form? What is the final form?

Fractional Form: x²+3x+5/x+1=x+2+(3/x+1) Final Form: x²+3x+5=(x+1)(x+2)+3

Graph of f(x)=ax², a<0 Domain? Range? Intercept? Increasing on? Decreasing on? Even or odd? Axis of symmetry? Relative maximum or vertex?

Graph of f(x)=ax², a<0 Domain: (-inf, inf) Range: (-inf, 0] Intercept: (0,0) Increasing on: (-inf, 0) Decreasing on: (0, inf) Even or odd: Even Axis of symmetry: x=0 Relative maximum or vertex: (0,0)

Graph of f(x)=ax², a>0 Domain? Range? Intercept? Decreasing on? Increasing on? Even or odd? Axis of symmetry? Relative minimum or vertex?

Graph of f(x)=ax², a>0 Domain: (-inf, inf) Range: [0, inf) Intercept: (0,0) Decreasing on: (-inf, 0) Increasing on: (0, inf) Even or odd: Even Axis of symmetry: x=0 Relative minimum or vertex: (0,0)

For a polynomial function, a factor of (x-a)^k, k>1, yields a repeated zero x=a of multiplicity k. If k is odd, then the graph...?

If k is odd, then the graph crosses the x-axis at x=a

What are the graphs of polynomial functions of degree 2?

Parabolas

What is the graph of the standard form of a quadratic function?

Standard Form: f(x)=a(x-h)²+k, a≠0 The graph of f is a parabola whose axis is the vertical line x=h and whose vertex is the point (h,k)

What does the coefficient a determine?

The coefficient a determines how widely the parabola given by f(x)=ax² opens.

In the standard form a+bi, what is the real number a called?

The real part of the complex number a+bi

How do you find the x-intercepts of the graph of f(x)=ax²+bx+c?

To find the x-intercepts of the graph of f(x)=ax²+bx+c, solve the equation ax²+bx+c=0. When ax²+bx+c does not factor, use the quadratic formula to find the intercepts

Two complex numbers a+bi and c+di are equal to each other if and only if...?

a=c and b=d

Write the polynomial f(x)=x⁴-x²-20 as the product of factors that are irreducible over the rationals

x⁴-x²-20=(x²-5)(x²+4)