# Recent questions in Polynomial graphs

Polynomial graphs

### Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: $$\displaystyle{P}{\left({x}\right)}={a}{\left({x}−α\right)}{\left({x}−β\right)}{\left({x}−γ\right)},{a}≠{0}$$ Type 2: Two real zeros, one repeated: $$\displaystyle{P}{\left({x}\right)}={a}{\left({x}−α\right)}{2}{\left({x}−β\right)},{a}≠{0}$$ Type 3: One real zero repeated three times: $$\displaystyle{P}{\left({x}\right)}={a}{\left({x}−α\right)}{3},{a}≠{0}$$ Type 4: One real and two imaginary zeros: $$\displaystyle{P}{\left({x}\right)}={\left({x}−α\right)}{\left({a}{x}{2}+{b}{x}+{c}\right)},Δ={b}{2}−{4}{a}{c}{<}{0},{a}≠{0}$$ Experiment with the graphs of Type 1 cubics. Clearly state the effect of changing both the size and sign of a. What is the geometrical significance of $$\alpha, \beta, and\ \gamma ? \alpha,\beta,and\ \gamma$$?

Polynomial graphs

### For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Double zero at $$x = 1$$, and triple zero at $$x = 3$$. Passes through the point (2, 15).

Polynomial graphs

### Copy and complete the anticipation guide in your notes. StatementThe quadratic formula can only be used when solving a quadratic equation.Cubic equations always have three real roots.The graph of a cubic function always passes through all four quadrants.The graphs of all polynomial functions must pass through at least two quadrants.The expression $$x2>4$$ is only true if $$x>2$$.If you know the instantaneous rates of change for a function at $$x = 2$$ and $$x = 3$$, you can predict fairly well what the function looks like in between. Agree Disagree Justification Statement The quadratic formula can only be used when solving a quadratic equation. Cubic equations always have three real roots. The graph of a cubic function always passes through all four quadrants. The graphs of all polynomial functions must pass through at least two quadrants. The expression $$x2 >4$$ is only true if $$x>2$$. If you know the instantaneous rates of change for a function at $$x = 2$$ and $$x = 3$$, you can predict fairly well what the function looks like in between.

Polynomial graphs

### For the following exercises, use the given information about the polynomial graph to write the equation. Degree 4. Root of multiplicity 2 at $$x = 4,$$ and roots of multiplicity 1 at $$x = 1$$ and $$x = minus$$;2. y-intercept at (0, minus;3).

Polynomial graphs

### For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at $$x = -5$$, $$x =-2$$, and $$x = 1$$. y-intercept at (0, 6)

Polynomial graphs

### How does one find the lovely asymptotes of a polynomial graph?

Polynomial graphs

### For the following exercise, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y-intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. $$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{2}}-{3}{x}-{5}$$

Polynomial graphs

### Find the quadratic polynomial whose graph passes through the points (0, 0), (-1, 1), and (1, 1).

Polynomial graphs

### Carla is using the Rational Root Theorem and synthetic division to find real roots of polynomial functions and sketch their graphs. She says she has wasted her time when she tests a possible root and finds a nonzero remainder instead. Do you agree with her statement? Explain your reasoning.

Polynomial graphs

### For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at $$x = -2,$$ $$x = 1$$, and $$x = 3$$. y-intercept at $$(0, -4)$$.

Polynomial graphs

### For the following exercise, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y-intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. $$\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}+{3}{x}^{{2}}-{x}-{3}$$

Polynomial graphs

### Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: $$P(x)=a(x- \alpha)(x− \beta)(x−γ),a \neq 0$$ Type 2: Two real zeros, one repeated: $$P(x)=a(x-\alpha)2(x−\beta),a \neq 0$$ Type 3: One real zero repeated three times: $$P(x)=a(x- \alpha)3,a\neq0$$ Type 4: One real and two imaginary zeros: $$P(x)=(x-\alpha)(ax2+bx+c),\triangle=b2−4ac<0,a \neq 0$$ Experiment with the graphs of Type 4 cubics. What is the geometrical significance of $$\alpha \alpha$$; and the quadratic factor which has imaginary zeros?

Polynomial graphs

### How can proficiency in factoring help when graphing polynomial functions without using a calculator?

Polynomial graphs

### Explain what you know about the roots of polynomial functions. How can you determine all of the roots? Give examples, including sketches of graphs.

Polynomial graphs

### The graph of a polynomial function has the following characteristics: a) Its domain and range are the set of all real numbers. b) There are turning points at $$x = -2$$, 0, and 3. a) Draw the graphs of two different polynomial functions that have these three characteristics. b) What additional characteristics would ensure that only one graph could be drawn?

Polynomial graphs

### For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at $$x = minus;3$$ and $$x = 2$$ and a root of multiplicity 1 at $$x=minus;2$$. y-intercept at (0, 4).

Polynomial graphs

### Solve the polynomial inequality graphically. $$\displaystyle−{x}^{{3}}−{3}{x}^{{2}}−{9}{x}+{4}{<}{0}$$

Polynomial graphs

### For the following exercise, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y-intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. $$\displaystyle{f{{\left({x}\right)}}}={3}{x}−{x}^{{3}}$$

Polynomial graphs

### Find a formula for the function described. A fourth-degree polynomial whose graph is symmetric about the y-axis, has a y-intercept of 0, and global maxima at (1, 2) and (-1, 2).

Polynomial graphs

...