Answers CMS 2026-2027 Foundations of Math 3 INITIAL CREDITPolynomials and Rational Functions

Polynomials and Rational Functions — Test Answers

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What is the product? the fraction with numerator 4 n and denominator 4 n minus 4 times the fraction with numerator n minus 1 and denominator n plus 1

the fraction with numerator 1 and denominator n plus 1

the fraction with numerator 4 and denominator n plus 1

the fraction with numerator n and denominator n plus 1

the fraction with numerator 4 n and denominator n plus 1

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Which expression is equivalent to the fraction with numerator m minus 4 and denominator m plus 4 divided by open paren m plus 2 close paren ?

the fraction with numerator open paren m plus 4 close paren times open paren m plus 2 close paren and denominator m minus 4

the fraction with numerator m minus 4 and denominator open paren m plus 4 close paren times open paren m plus 2 close paren

the fraction with numerator m plus 4 and denominator open paren m minus 4 close paren times open paren m plus 2 close paren

the fraction with numerator open paren m minus 4 close paren times open paren m plus 2 close paren and denominator m plus 4

How can Ari simplify the following expression? the fraction with numerator the fraction with numerator 5 and denominator A minus 3 minus 4 and denominator 2 plus the fraction with numerator 1 and denominator A minus 3

Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerators and multiply the denominators.

Divide the numerator and the denominator by A minus 3 . Then simplify the numerator and simplify the denominator.

Write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator.

Divide the numerator and the denominator by A minus 3 . Then divide the numerator by the denominator.

Which expression is equivalent to the quotient? the fraction with numerator the fraction with numerator m plus 3 and denominator m squared minus 16 and denominator the fraction with numerator m squared minus 9 and denominator m plus 4

the fraction with numerator 1 and denominator open paren m plus 4 close paren times open paren m plus 3 close paren

the fraction with numerator 1 and denominator open paren m minus 4 close paren times open paren m minus 3 close paren

the fraction with numerator m plus 3 and denominator m plus 4

the fraction with numerator m minus 4 and denominator m minus 3

The owner of a coffee shop chain creates a graph representing sales in thousands of dollars. She describes the characteristics as follows: • There have been two turning points in sales this year. • Sales are expected to increase in the long run. • The least amount of sales during the year is about 250 dollars comma 000 Which graph correctly shows the coffee shop’s sales?

Image with description Line graph on an x y coordinate plane. The x-axis is labeled from 1 to 9. The y-axis is labeled from 100 to 900 in increments of 100. The curve begins at approximately (0 comma 300), rises to a local maximum at (1 point 5 comma 525), then decreases to a local minimum near (5 comma 250), and subsequently increases sharply to approximately (10 comma 800).

Image with description Line graph on an x y coordinate plane. The x-axis is labeled from 1 to 9. The y-axis is labeled from 100 to 900 in increments of 100. The curve begins at approximately (0 comma 200), rises to a local maximum at (1 point 5 comma 425), then decreases to a local minimum near (5 point 5 comma 150), and subsequently increases sharply to approximately (10 comma 700).

Image with description Line graph on an x y coordinate plane. The x-axis is labeled from 1 to 9. The y-axis is labeled from 100 to 900 in increments of 100. The curve begins near (0 comma 450), decreases to a local minimum at approximately (1 point 5 comma 250), then increases to a local maximum at approximately (5 point 5 comma 525), and finally decreases sharply, passing through (9 comma 50) and continuing downwards.

Image with description Line graph on an x y coordinate plane. The x-axis is labeled from 1 to 9. The y-axis is labeled from 100 to 900 in increments of 100. The curve begins near (0 comma 425), decreases to a local minimum at approximately (3 point 5 comma 250), then rises sharply, ending with an arrow near (10, 900) indicating an upward path.

Which graph represents the function f of x is equal to 4 over x ?

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches the x-axis as x approaches negative infinity from the right and approaches the y-axis as x approaches 0 from the left. Its turning point is near (negative 2 comma negative 2). The branch on the right approaches the y-axis as x approaches 0 from the right and approaches the x-axis as x approaches infinity from the left. Its turning point is near (2 comma 2).

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches the x-axis as x approaches negative infinity from the right and approaches the y-axis as x approaches 0 from the left. Its turning point is near (negative 1 comma 1). The branch on the right approaches the y-axis as x approaches 0 from the right and approaches the x-axis as x approaches infinity from the left. Its turning point is near (1 comma negative 1).

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches the x-axis as x approaches negative infinity from the right and approaches the y-axis as x approaches 0 from the left. Its turning point is near (negative 2 comma 2). The branch on the right approaches the y-axis as x approaches 0 from the right and approaches the x-axis as x approaches infinity from the left. Its turning point is near (2 comma negative 2).

Image with description A line graph on a Cartesian coordinate plane shows a rational function with two separate branches. The x-axis and y-axis are labeled with even integer values from negative 8 to 8. One branch lies in the first quadrant, passing through (1 comma 1) and approaching both axes without touching them. The second branch lies in the third quadrant, passing through (negative 1 comma negative 1) and also approaching both axes asymptotically.

The inverse variation equation shows the relationship between wavelength in meters, x , and frequency, y y is equal to the fraction with numerator 3 times 10 to the 8th power and denominator x What is the wavelength for radio waves with frequency 3 times 10 to the 9th power ?

3 times 10 to the 17th power m

3 times 10 inverse m

9 times 10 to the 17th power m

1 times 10 inverse m

A polynomial function has a root of negative 3 with multiplicity 2 , a root of 0 with multiplicity 1 , a root of 1 with multiplicity 1 , and a root of 3 with multiplicity 2 If the function has a positive leading coefficient and is of even degree, which could be the graph of the function?

Image with description On a coordinate plane, the x and y axes ranges from negative 8 to 8 with an increment of 1. The polynomial graph is drawn on the coordinate plane. It starts from quadrant I passing through the point negative 3 point 8 comma 8 crosses x axis at negative 3 comma 0, goes down and curves up at the point negative 2 comma negative 3 point 5, crosses the y axis at the point 0 comma negative 0 point 8. Here the curve touches the x axis at point 1 comma 0, goes down slightly and again crosses the x axis at point 3 comma 0, and moves to quadrant I through the point 4 comma 7.

Image with description On a coordinate plane, the x and y axes ranges from negative 8 to 8 with an increment of 1. The polynomial graph is drawn on the coordinate plane. The curve starts from negative 3 point 8 comma 8, touches the x axis at negative 3 comma 0, passes through the point negative 1 point 5 comma 3 point 5 and comes down to the origin. Here the curve goes down slightly and curves up to the point 2 point 1, curves down to touch x axis at point 2 comma 0 and moves to quadrant I through the point 4 comma 8.

Image with description On a coordinate plane, the x and y axes ranges from negative 8 to 8 with an increment of 1. The polynomial graph is drawn on the coordinate plane. The curve starts from quadrant III, passing through the point negative 3 point 5 comma negative 8, touches the x axis at 3 comma 0, moves down and curves up at negative 1 point 8 comma 7, and passes through the origin. Here the curve moves up slightly and passes through the point 2 comma 0, goes down, curves up to the point 2 comma negative 2, crosses x axis once again at point 3 comma 0 and moves to quadrant IV through the point 3 point 5 comma negative 8.

Image with description On a coordinate plane with x and y axes marked from negative 8 to 8, a polynomial graph is drawn. The graph starts from quadrant 3 passing through the point (negative 3 point 5 comma negative 8), it crosses the x axis at (negative 3 comma 0) moves up and curves down at (negative 2 comma 7), crosses the y axis at (0 comma 1), crosses the x axis at (1 comma 0), moves up and curves down at (2 point 5 comma 1). It then moves down, crosses x axis at (2 comma 0) and moves to quadrant 4 through the point (4 comma negative 8).

Which statement describes how the graph of the given polynomial would change if the term negative 3 x to the 6th power is added? y is equal to 2 x to the 6th power plus 9 x to the 5th power minus 7 x cubed minus 1

Both ends of the graph will approach negative infinity.

Both ends of the graph will approach positive infinity.

The ends of the graph will approach zero.

The ends of the graph will extend in opposite directions.

Where are the asymptotes for the following function located? f of x is equal to the fraction with numerator 7 and denominator x squared minus 2 x minus 24

x is equal to negative 4 and x is equal to 6

x is equal to 4 and x is equal to negative 6

x is equal to negative 4 and x is equal to 7

x is equal to 6 and x is equal to 7

A polynomial function has a root of negative 6 with multiplicity 1 , a root of negative 2 with multiplicity 3 , a root of 0 with multiplicity 2 , and a root of 4 with multiplicity 3 If the function has a positive leading coefficient and is of odd degree, which statement about the graph is true?

The graph of the function is negative on open paren negative infinity comma 0 close paren .

The graph of the function is positive on open paren negative 6 comma negative 2 close paren .

The graph of the function is negative on open paren 4 comma infinity close paren .

The graph of the function is positive on open paren negative 2 comma 4 close paren .

Using cap v is equal to l w h , what is an expression for the volume of the following prism? Open image in full-screen view

the fraction with numerator 4 and denominator 3 d minus 12

the fraction with numerator 4 d minus 8 and denominator 3 times open paren d minus 4 close paren squared

the fraction with numerator 4 times open paren d minus 2 close paren and denominator 3 open paren d minus 3 close paren open paren d minus 4 close paren

the fraction with numerator 1 and denominator 3 d minus 3

Which expression is equivalent to the following complex fraction? the fraction with numerator negative 2 over x plus 5 over y and denominator 3 over y minus 2 over x

the fraction with numerator x squared y squared and denominator open paren negative 2 y plus 5 x close paren times open paren 3 x minus 2 y close paren

the fraction with numerator open paren negative 2 y plus 5 x close paren times open paren 3 x minus 2 y close paren and denominator x squared y squared

the fraction with numerator negative 2 y plus 5 x and denominator 3 x minus 2 y

the fraction with numerator 3 x minus 2 y and denominator negative 2 y plus 5 x

How does the graph of g of x is equal to the fraction with numerator 1 and denominator x plus 4 minus 6 compare to the graph of the parent function f of x is equal to 1 over x ?

g of x is shifted 4 units right and 6 units down from f of x .

g of x is shifted 4 units left and 6 units up from f of x .

g of x is shifted 4 units right and 6 units up from f of x .

g of x is shifted 4 units left and 6 units down from f of x .

Which statement is true about the discontinuities of the function f of x ? f of x is equal to the fraction with numerator x plus 1 and denominator 6 x squared minus 7 x minus 3

There are holes at x is equal to 3 halves and x is equal to negative 1 third .

There are asymptotes at x is equal to 3 halves and x is equal to negative 1 third .

There are holes at x is equal to negative 3 halves and x is equal to 1 third .

There are asymptotes at x is equal to negative 3 halves and x is equal to 1 third .

The increase in a person’s body temperature cap t times t , above 98.6 degrees F, can be modeled by the function cap t times t is equal to the fraction with numerator 4 t and denominator t squared plus 1 , where t represents time elapsed. What is the meaning of the horizontal asymptote for this function?

The horizontal asymptote of y is equal to 4 means that the person’s temperature will approach 102.6 degrees F as time elapses.

The horizontal asymptote of y is equal to 4 means that the person’s temperature will approach 4 degrees F as time elapses.

The horizontal asymptote of y is equal to 0 means that the person’s temperature will approach 0 degrees F as time elapses.

The horizontal asymptote of y is equal to 0 means that the person’s temperature will approach 98.6 degrees F as time elapses.

The equation cap d is equal to 13 over cap v shows that the density of a particular substance equals a mass of 13 grams divided by the volume of the substance. What happens to the density as the volume approaches 0 ?

The density approaches 1 .

The density approaches 13 .

The density approaches infinity.

The density approaches 0 .

Which graph represents the function f of x is equal to the fraction with numerator 2 x and denominator x squared minus 1 ?

Image with description A coordinate plane with a graph showing three branches. The branch on the left approaches the x-axis as x approaches negative infinity from the right and approaches negative infinity as x approaches negative 1 from the left. The branch in the middle approaches infinity as x approaches negative 1 from the right and approaches negative infinity as x approaches 1 from the left. It has a turning point at the origin. The branch on the right approaches infinity as x approaches 1 from the right and approaches the x-axis as x approaches infinity from the left.

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches y equals 1 as x approaches negative infinity from the right and approaches negative infinity as x approaches 0 from the left. The branch on the right approaches negative infinity as x approaches 1 from the right and approaches y equals 1 as x approaches infinity from the left.

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches y equals 2 as x approaches negative infinity from the right and approaches negative infinity as x approaches 1 from the left. The branch on the right approaches infinity as x approaches 1 from the right and approaches y equals 2 as x approaches infinity from the left.

Image with description A coordinate plane with a graph showing three branches. The branch on the left approaches y equals 2 as x approaches negative infinity from the right and approaches infinity as x approaches negative 1 from the left. The branch in the middle approaches negative infinity as x approaches negative 1 from the right and approaches negative infinity as x approaches 1 from the left. The branch on the right approaches infinity as x approaches 1 from the right and approaches y equals 2 as x approaches infinity from the left.

What is the difference? the fraction with numerator x and denominator x squared minus 16 minus the fraction with numerator 3 and denominator x minus 4

the fraction with numerator x minus 3 and denominator open paren x plus 5 close paren times open paren x minus 4 close paren

the fraction with numerator negative 2 times open paren x minus 6 close paren and denominator open paren x plus 4 close paren times open paren x minus 4 close paren

the fraction with numerator negative 2 times open paren x plus 6 close paren and denominator open paren x plus 4 close paren times open paren x minus 4 close paren

the fraction with numerator 2 times open paren x plus 6 close paren and denominator open paren x plus 4 close paren times open paren x minus 4 close paren

Mr. Knotts found the difference of the following expression. Which statement is true about Mr. Knotts’s work? the fraction with numerator x and denominator x squared minus 1 minus the fraction with numerator 1 and denominator x minus 1 Knotts’s Work Step 1: the fraction with numerator x and denominator open paren x plus 1 close paren times open paren x minus 1 close paren minus the fraction with numerator 1 and denominator x minus 1 Step 2: the fraction with numerator x and denominator open paren x plus 1 close paren times open paren x minus 1 close paren minus the fraction with numerator 1 times open paren x plus 1 close paren and denominator open paren x plus 1 close paren times open paren x minus 1 close paren Step 3: the fraction with numerator x minus x plus 1 and denominator open paren x plus 1 close paren times open paren x minus 1 close paren Step 4: the fraction with numerator 1 and denominator open paren x plus 1 close paren times open paren x minus 1 close paren

In step 1, he did not correctly factor the denominator.

In step 4, he did not correctly subtract the numerator.

In step 2, he did not use the correct common denominator.

In step 3, he did not distribute the negative sign when he subtracted.

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Polynomials and Rational Functions — Test Answers

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