Adding and Subtracting Rational Expressions

We are asked to find a simplified form of the thin-lens equation. When we look at the choices, we see that the variable f is on the left side for each equation, so our first step will be to solve the thin-lens equation for f.

The result corresponds to option D.

We found a simplified equation for variable f in Part A. f = d_i d_o/d_o+d_i We can use this equation to find the focal length f of a lens when we know the actual distance d_o of the object from the lens and the distance d_i of the image from the lens. Let's show what we know on a diagram.

We can substitute d_o=50 and d_i=20 into our equation and find the focal length of the lens.

The focal length is about 14.3 centimeters.

Numbers	Multiples of Numbers	Common Multiples	Least Common Multiple
$2$ and $3$	$Multiples of 2 : Multiples of 3 : 2, 4, 6, 8, 10, 12, \dots 3, 6, 9, 12, 15, \dots$	$\underline{6}, 12, 18, 24, \dots$	$LCM (2, 3) = 6$
$8$ and $12$	$Multiples of 8 : Multiples of 12 : 8, 16, 24, 32, 40, 48, \dots 12, 24, 36, 48, \dots$	$\underline{24}, 48, 72, 96, \dots$	$LCM (8, 12) = 24$

Coprimes	LCM
$3$ and $5$	$15$
$5$ and $4$	$20$
$4$ and $9$	$36$

Polynomials	Factor	LCM	Explanation
$4 x^{3} and 6 x y^{2}$	$2^{2} \cdot x^{3} and 2 \cdot 3 \cdot x \cdot y^{2}$	$12 x^{3} y^{2}$	$12 x^{3} y^{2}$ is the smallest expression that is divisible by both $4 x^{3}$ and $6 x y^{2} .$
$3 x^{3} + 3 x^{2} and x^{2} + 5 x + 4$	$3 \cdot x^{2} \cdot (x + 1) and (x + 1) (x + 4)$	$3 x^{2} (x + 1) (x + 4)$	$3 x^{2} (x + 1) (x + 4)$ is the smallest expression that is divisible by both $3 x^{3} + 3 x^{2}$ and $x^{2} + 5 x + 4 .$

	Standard Form	Factored Form	All Related Factors
Polynomail I	$12 x^{2} y + 48 x y + 48 y$	$2^{2} \cdot 3 \cdot y \cdot (x + 2)^{2}$	$2^{2} \cdot 3^{1} \cdot x^{0} \cdot y^{1} \cdot (x + 2)^{2} \cdot (x - 8)^{1}$
Polynomail II	$3 x^{3} y - 18 x^{2} y - 48 x y$	$3 \cdot x \cdot y \cdot (x + 2) \cdot (x - 8)$	$2^{0} \cdot 3^{1} \cdot x^{1} \cdot y^{1} \cdot (x + 2)^{1} \cdot (x - 8)^{1}$

	Given Form	Factored Form	All Related Factors
Heichi	$6 x^{2} y - 6 x$	$2 \cdot 3 \cdot x \cdot (x y - 1)$	$2^{1} \cdot 3^{1} \cdot x^{1} \cdot (x y - 1)^{1}$
Ramsha	$3 x^{3} y^{2} - 6 x^{2} y + 3 x$	$3 \cdot x \cdot (x y - 1)^{2}$	$2^{0} \cdot 3^{1} \cdot x^{1} \cdot (x y - 1)^{2}$

Adding and Subtracting Rational Expressions

Catch-Up and Review

Egyptian Fractions

Least Common Multiple of Polynomials and How to Find It

Least Common Multiple

Finding the Least Common Multiple of Polynomials

Writing an Expression for the Meeting Time

Hint

Solution

Using the Least Common Denominator to Add or Subtract Rational Expressions

Least Common Denominator