Month: March 2024 (Page 2 of 3)

Announcements

WebWork schedule:

• Rational Expressions-Complex Fractions 2 – due Fri March 22 (Sec 1.3.4)
• Radical Expressions-Adding and Subtracting – due Fri March 22 (Sec 1.4.4)
• Radical Expressions-Multiplying – due Mon March 25 (Sec 1.4.4)

Topics

We continued studying “radical expressions”–we briefly discussed “higher roots” (such as cube roots, fourth roots, etc.) and now these can be expressed as “rational exponents”:

We then revisited factoring a quadratic polynomial, and how we can use a factorization to solve a quadratic equation. We can also solve such a quadratic equation using the quadratic formula (which involves a square root):

But for a quadratic polynomial that doesn’t factor, we need to use the quadratic formula–and then we usually can simplify the square root involved, using the techniques we have been studying:

We also looked at the graphs of these quadratic polynomials in Desmos, and noticed how the solutions of the quadratic equation correspond to the x-intercepts of the graph.

We finished by going through some more WebWork exercises on adding/subtracting and then multiplying square roots:

Note that exercises above involve the distributive property and FOIL.

Announcements

WebWork schedule:

• Rational Expressions-Complex Fractions 1 – due Mon March 18 (Sec 1.3.4)
• Radical Expressions-Simplifying – due Mon March 18 (Sec 1.4.2)
• Rational Expressions-Complex Fractions 2 – due Wed March 20 (Sec 1.3.4)
• Radical Expressions-Adding and Subtracting – due Fri March 22 (Sec 1.4.4)

Topics

We studied “radical expressions”, i.e., expressions involving square roots, and went through another complex fractions example.

We started by discussing what the square root of a number is:

In order to simplify square root expressions, we primarily use the following “product property” of square roots (“the square root of a product is the product of the square roots”):

(However, note that this does not work for the square root of a sum, as we can see from simple examples:

We can use the product property of square roots to simplify square root expressions–the strategy is to look for “perfect square factors” of the number (or the expression) inside the square root:

We use the same strategy for square roots involving algebraic expressions:

We did some exercises from the WebWork using this technique:

We can also simplify square root expressions involving ratios, using a similar property:

We then introduced how to add/subtract square root expressions–here we treat square roots of the same number/expression as “like terms”:

Finally, we returned to complex fractions, and did an example from “Complex Fractions Part 2”, which involve multiple steps:

Announcements

WebWork schedule:

• Rational Expressions-Complex Fractions 1 – due Mon March 18 (Sec 1.3.4)
• Radical Expressions-Simplifying – due Mon March 18 (Sec 1.4.2)
• Rational Expressions-Complex Fractions 2 – due Wed March 20 (Sec 1.3.4)

Topics

We reviewed an exercise from Exam #1, on factoring quadratic polynomials, and showed how we can use that to solve quadratic equations. We also previewed the quadratic formula, which used square roots–our next topic will be square roots and other “radical expressions.”

We also returned to rational expressions, to introduced “complex fractions” (a fraction that has fractions within its numerator and/or denominator):

Exercise from Exam #1, and using the factored polynomial to solve a quadratic equation:

We can also solve such quadratic equations using the quadratic formula–which we will cover in more depth in a few weeks. Note that the quadratic formula involves a square root:

We also looked at another factoring exercise from Exam #1, and looked an associated quadratic equation, which we can solve using the factoring, the quadratic formula–or in this case, we can solve it “directly”, using square roots:

Finally, we looked at complex fractions: