MrE's Algebra Blog

Algebra: Chapter 5, Lesson 5, page 221

Polynomials

Polynomials is the catch term for monomials put together with `+` or `–` signs. Polynomials with just 1 term are called “monomials”, with 2 terms they are called “binomials”, with 3 terms, they are called “trinomials”. Polynomials with more than 3 terms have no particular name.

TERMS are separated by `+` or `–` signs and the FACTORS are the things that are multiplied together to get each term. The numeric factor of a term is called a COEFFICIENT and terms with just numbers (no variables) are called CONSTANTS.

The DEGREE (or ORDER) of a term is the sum of the exponents of the variables and the degree of a polynomial is the highest degree of its terms. The term with the highest degree is called the LEADING TERM and the coefficient of the leading term is called the LEADING COEFFICIENT.

We can simplify a polynomial by collecting LIKE TERMS. Like terms MUST have the same variables in the terms AND must have the same exponent values, this part is important.

Examples:

`2m^3 − 6m^3=(2−6)m^3=−4m^3`
`5x^3 + 6x^3 + 4 = 11x^3 + 4`

Click here (there are 2 pages) for some examples.

Two of tonight’s homework problems solved by MrE are here! Just click it!

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Algebra 1a: Chapter 4, Lesson 4, page 183.

Using the Properties Together

Inequalities of 2 steps
We do the same, again, as equalities and solve by:

• Distributing when required
• Combining like terms
• Adding or subtracting terms to isolate variables and numbers (constants)
• Multiplying or dividing to finish solving for the variable, remembering to reverse the sign of the inequality IF we multiply or divide by a NEGATIVE NUMBER.

For example:

`7x + 4 ≤ 4x + 16`

subtract `4x` from both sides, that looks like

`7x – 4x + 4 ≤ 4x – 4x + 16`

now combine like terms on the left and the right sides

`3x + 4 ≤ 16`

subtract 4 from each side

`3x + 4 – 4 ≤ 16 – 4`

combine like terms again on both sides, so that

`3x ≤ 12`

and finally divide both sides by 3

`(3x)/3 ≤ 12/3`, so that finally

`x ≤ 4`

Go slow and show all the steps! Here are some more examples from purplemath.com

Two of tonight’s homework problems for Lesson 4 solved by MrE are here! Just click it!

Algebra: Chapter 5, Lesson 4, page 217.

Scientific Notation

We can write numbers as the product of a power of 10 and a number greater than 1 but less than 10 (9.9999…. to be exact). Standard notation is the stuff that we are normally used to.

Examples: `4.58 × 10^4=45,800`
`(3.0 × 10^5)(4.1 × 10^(-3))=(3.0⋅4.1 ) * (10^5 × 10^(-3))=12.3 × 10^2=1.23 × 10^3`

`(2.5 × 10^(-7))/(5.0 × 10^6) = (2.5/5.0) * (10^(-7)/10^6) = 0.5 × 10^(-13) = 5.0 × 10^(-14)`

More scientific notation examples are here.

Two of tonight’s homework problems solved by MrE are here! Just click it!

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Algebra 1a: Chapter 4, Lesson 3, page 180.

The Multiplication Property of Inequalities

The property states if `c` is POSITIVE

• if `a < b`, then `ac < bc` and
• if `a > b`, then `ac > bc`

Where `c` is NEGATIVE

• if `a < b`, then `ac > bc` and
• if `a > b`, then `ac < bc`

Following the EXACT same steps as equalities, we have learned to solve 1 step equations with inequalities. The ONLY difference is when multiplying or dividing by a NEGATIVE number, we must REVERSE the sign of the inequality for the final solution!! If we divide or multiply by a positive number, we leave the inequality sign alone.

Here are some examples from purplemath.com that have to do with inequalities with products and divisions.

Two of tonight’s homework problems are here for Chapter 4-3 as well!

Algebra: Geometry and Nature

Geometric Snowflake Projects! Nicely done!! Enjoy the winter break!!!

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Algebra 1a: Geometry and Nature

Geometric Snowflake Projects! Nicely done!! Enjoy the winter break!!!

Algebra: Chapter 5, Lesson 3, page 214.

Multiplying and Dividing Monomials

A monomial is an expression that is either a NUMERAL, an VARIABLE or a PRODUCT of numerals and variables with whole number exponents. If the monomial is a numeral, we call it a CONSTANT.

Using the properties we had from yesterday, we can use the associative and commutative properties to multiply or divide monomials.

For example, `(3x)(4x)=(3⋅4⋅x⋅x)=12x^2` or

`(x^5)(x^-2) =x^(5-2)=x^3`

Here are some more examples too!

Two of tonight’s homework problems solved by MrE are here! Just click it!

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Algebra 1a: Chapter 4, Lesson  3, page 180.

The Multiplication Property of Inequalities

The property states if `c` is POSITIVE

• if `a < b`, then `ac < bc` and
• if `a > b`, then `ac > bc`

Where `c` is NEGATIVE

• if `a < b`, then `ac > bc` and
• if `a > b`, then `ac < bc`

Following the EXACT same steps as equalities, we have learned to solve 1 step equations with inequalities. The ONLY difference is when multiplying or dividing by a NEGATIVE number, we must REVERSE the sign of the inequality for the final solution!! If we divide or multiply by a positive number, we leave the inequality sign alone.

Here are some examples from purplemath.com that have to do with inequalities with products and divisions.

Two of tonight’s homework problems are solved by MrE  for Chapter 4-3 as well!

Algebra: Chapter 5, Lesson 2, page 209.

Exponents and More!

Exponents and More with Exponents

For like bases, we have:

1. Rule: `a^0=1`
2. Rule: to multiply we do `a^m⋅a^n=a^(m+n)
3. Rule: to divide, we do `a^m/a^n=a^(m-n)`
4. Rule: for negative exponent, we can express them as positive by, `a^(-m)=1/a^m
5. Rule: for raising a power to another power, `(a^m)^n=a^(mn)`
6. Rule: for raising a product to a power, `(ab)^n=a^n⋅b^n`
7. Rule: for raising a quotient to a power, `(a/b)^n=a^n/b^n`

Remember and MEMORIZE THESE RULES for Lessons 1 and 2. Practice here!

PurpleMath has an EXCELLENT 2 page tutorial, click here!

Two of tonight’s homework problems solved by MrE are here! Just click it.

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Algebra 1a: Chapter 4, Lesson 2, page 175.

The Addition Property of Inequalities

Don’t forget, the equations:

• if `a < b`, then `a + c < b + c`
• if `a > b`, then `a + c > b + c`

and similar statements are true for ≤ and ≥

For inequality of one step, follow the EXACT same steps as equalities. The only things we have to remember when graphing on a number line:

• For the symbols ≤ and ≥, the circle must be CLOSED because we INCLUDE the data point
• For the symbols < and >, the circle must be OPEN because we get as close as possible to the data point but it is NOT INCLUDED!

Here is a link from PURPLEMATH.com with some more examples!

SNOW DAY #1!

Algebra: Chapter 5, Lesson 1, page 204.

Exponents!

Exponents and More with Exponents

Remember, an exponent tells how many times we use a base as a factor. For example, `a^3=a⋅a⋅a`. An expression written with exponents is written using exponential notation.

For like bases, we have:

1. Rule: `a^0=1`
2. Rule: to multiply we do `a^m⋅a^n=a^(m+n)
3. Rule: to divide, we do `a^m/a^n=a^(m-n)`
4. Rule: for negative exponent, we can express them as positive by, `a^(-m)=1/a^m
5. Rule: for raising a power to another power, `(a^m)^n=a^(mn)`
6. Rule: for raising a product to a power, `(ab)^n=a^n⋅b^n`
7. Rule: for raising a quotient to a power, `(a/b)^n=a^n/b^n`

Remember and MEMORIZE THESE RULES for Lessons 1 and 2. Practice here!

They are also included in the Algebra Cheat Sheet that was passed today. As long as you remember these 7 formulae, you’ll be OK. Remember, if you forget the rules, just write out the problem and see what can be simplified!

PurpleMath has an EXCELLENT 2 page tutorial, click here!

Two of tonight’s homework problems solved by MrE are here! Just click it.

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Algebra 1a: Chapter 4, Lesson 1, page 172.

Inequalities and Their Graphs

Symbols:

• `<`, is less than
• `≤`, is less than or equal to
• `>`, is greater than
• `≥`, is greater than or equal to

Open or closed circles on the number line:

• OPEN CIRCLE  (get as close to the value BUT don’t touch) it for `<` or `>`
• CLOSED CIRCLE (included the value on the number line) for `≤` or `≥`

Algebra: Chapter 9 Benchmark (#7)

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Algebra 1a: Chapter 4, Lesson 1, page 172.

Inequalities and Their Graphs

Symbols:

• `<`, is less than
• `≤`, is less than or equal to
• `>`, is greater than
• `≥`, is greater than or equal to

Open or closed circles on the number line:

• OPEN CIRCLE  (get as close to the value BUT don’t touch) it for `<` or `>`
• CLOSED CIRCLE (included the value on the number line) for `≤` or `≥`

Algebra: Chapter 9, Review

Remember! IN GRAPHING SYSTEMS OF LINEAR INEQUALITIES! For the first equation, we can use any technique we learned from Chapter 7. Usually, the slope-intercept or `x` and `y` intercepts can be used to quickly define the line. Check a simple point like (0, 0) to see if that part of the ½ plane is true. If so, then shade that area.

Do the same for the other inequality and shade the appropriate ½ plane. The IMPORTANT PART is WHERE THE 2 INEQUALITIES OVERLAP THEIR SHADING, IS THE SOLUTION TO BOTH INEQUALITIES.

Have your notes organized for TOMORROW’s CHAPTER 9 TEST!

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Algebra 1a: Chapter 3, Benchmark #3

No homework!