Polynomial Identities

Review Topic 3 Polynomial Functions

Multiple Choice

For  $$x^3+2x^2+7x^4−3$$  , which of the following statements is true?

• The degree of the polynomial is 3.

• Written in standard form, the polynomial is  $$7x^4+x^3+2x^2−3.$$  

• The polynomial is a binomial.

• The leading coefficient is 1.

Multiple Choice

The graph of function f is shown. Use the zeros and the turning points of the graph to find the rule for f.

• $$f\left(x\right)=x^3-0.5$$

• $$f\left(x\right)=x^3-1$$

• $$f\left(x\right)=x^3+2x^2-x+\frac{1}{2\ }$$

• $$\ f\left(x\right)=x^3+\frac{1}{2}x^2-x$$

Multiple Choice

Simplify $$(12x^3–8x^2)\ –\ (3x^3+9x^2+x\ –\ 7)$$  .

• $$9x^3+x^2\ –\ x+7$$  

• $$9x^3-17x^2\ –\ x+7$$  

• $$15x^3-17x^2\ –\ x-7$$  

• $$15x^3+x^2\ +\ x-7$$  

Multiple Choice

$$(x^2+3x)(x^2-x+4)$$  Simplify .

• $$x^4-x^3+4x^2+12x$$  

• $$x^4+2x^3+7x^2-12x$$  

• $$x^4+2x^3+x^2+12x$$  

• $$x^4+2x^3-x^2+12x$$  

Multiple Choice

$$x^3+x^2−17x+15$$  

Write P(x) = as a product of two factors, using x − 3 as one factor.

• $$(x−3)(x^2+x−17)$$  

• $$(x−3)(x^2+4x−17)$$  

• $$(x−3)(x^2–4x−5)$$  

• $$(x−3)(x^2+4x−5)$$  

Multiple Choice

Use synthetic division to divide $$2x^4-3x^3+2x^2-8x-1$$   by x – 1.

• $$2x^3-5x^2-3x-11+\frac{23}{x-1}$$  

• $$2x^3-x^2+3x-5-\frac{6}{x-1}$$  

• $$2x^3-x^2+x-9-\frac{10}{x-1}$$  

• $$2x^3-x^2+x-7-\frac{8}{x-1}$$  

For the remainder. it the last number you solve for in the synthetic division chart.

Multiple Choice

What is the remainder when  $$f(x)=2x^4+x^3−8x−1$$   is divided by x − 2?

• -23

• 23

• -3

• 3

Use DESMOS

Go to desmos.com to help you solve the next problem. Match the graph with desmos graph.

Vaya a desmos.com para ayudarlo a resolver el próximo problema. Haga coincidir el gráfico con el gráfico desmos.

Multiple Choice

$$f(x)=x^3-2x^2-3x.$$  

Choose a possible sketch of the graph of 

Multiple Choice

Find the zeros of the function  $$f(x)=x^3−2x^2−19x+20$$   , and describe the behavior of the graph at each zero.

• The graph touches the x-axis at 4, and it crosses the x-axis at −1 and −5.

• The graph touches the x-axis at 4, −5, and 1.

• The graph touches the x-axis at 4, 5, and 1.

• The graph touches the x-axis at −4, and it crosses the x-axis at 1 and 5.

Multiple Choice

Choose the list that contains all of the possible rational solutions of  $$2x^4+x^3-x+7=0.$$  

• $$1,\ 7,\ \frac{1}{2},\ \frac{7}{2}$$  

• $$1,\ –1,\ 7,\ –7,\ \frac{1}{2},\ -\frac{1}{2},\ \frac{2}{7},\ -\frac{2}{7},\frac{7}{\ 2}\ ,-\frac{7}{2}$$  

• $$1,\ –1,\ 7,\ –7,\ \frac{1}{2},\ -\frac{1}{2},\ \frac{7}{\ 2}\ ,-\frac{7}{2}$$  

• $$0,\ 1,\ –1,\ 7,\ –7,\ \frac{1}{2},\ -\frac{1}{2},\frac{7}{\ 2}\ ,-\frac{7}{2}$$  

Multiple Choice

How does the graph of the function  $$f(x)=x^4−3$$   differ from the graph of its parent function?

• The graphs are the same.

• Subtracting 3 translates the graph down 3 units.

• The leading coefficient, 1, stretches the graph vertically.

• Subtracting 3 translates the graph up 3 units.