MrE's Algebra Blog

Algebra: Chapter 5 –  200 POINT BENCHMARK

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Algebra 1a: Chapter 7, Lesson 2, page 309.

Graphing Equations

We can graph an equation, by building a T-chart of values for both x and y. You can choose any values for `x ` and `y` when making your T-chart. I like to use values like 0, 1, and 2. Make them easy and try to pick AT LEAST 3 points when graphing and equation. YOU MUST USE A RULER WHEN CONNECTING THE DOTS TOO!

Sometimes, it can be easier when building the T-chart to “solve for `y`” first, this just cuts down on the workload. Solving for `y` means isolating the `y` variable to one side of the equation and keeping the constants and ALL other variables on the other side.

Here are 2 good links from purplemath.com, the first about graphing in general (lesson 1) and the second about the T-charts and lesson 2!

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

Algebra: Chapter 5 –  200 POINT BENCHMARK REVIEW SUMMARY

Addition of Polynomials

When we add polynomials, it can be easier if we order them in descending order for one of the variables. We can add polynomials in 2 ways:

1. Horizontally – scan across the polynomials and see what terms can be combined. REMEMBER, we can add terms ONLY IF they have the SAME VARIABLES and the SAME EXPONENTS.
2. Vertically – leave space for terms that are missing when making the columns, see the examples in the textbook on pagee 232.

Subtraction of Polynomials

We do just like the addition of polynomials, except for subtraction, we (remember way back to subtracting with negative numbers) add the opposite. The other thing we have to remember is the sign problems for double negatives, make “Change-change” or “bling-bling” with 2 negatives.

You can do the subtraction horizontally or vertically. Horizontally requires that you scan across the polynomials. It’s easier for me to put the polynomials in descending order and then combine like terms, remembering the combining terms MUST HAVE THE SAME VARIABLE AND THE SAME EXPONENTS!

If you do the problems horizontally, remember that the `-` sign turns everything in 2nd polynomial’s parenthesis `( x^2 …)` to its opposite sign!

Vertically (or columns in the textbook) require you to write out the problem with spaces for missing terms. It is easier to line up terms this way and you can stick in other terms as well that are not common to both polynomials.

Multiplying Polynomials

To multiply a monomial and a polynomial: multiply each term of the polynomial by the monomial

There are 3 techniques to multiply binomials:

1. FOIL (FIRST, OUTSIDE, INSIDE, LAST – or its derivatives for trinomials)
   • Multiply each term of a polynomial by EVERY OTHER TERM of the other polynomial
2. The BOX method
   • The BOX method, more like a rectangle with each term representing 1 side of an inner box. A binomial multiplied with a trinomial will be a BOX containing 2 x 3 number of smaller boxes inside it. Each term represents 1 edge in distance in the inner boxes.
3. The old fashioned multiplication method outlined on page 249.

You get to chose which is most comfortable for you BUT REMEMBER THE BOX for Chapter 6!

Remember too, the shortcuts for special binomials:

• `(A+B)(A+B)=(A+B)^2=A^2+B^2+2AB`
• `(A−B)(A−B)=(A−B)2=A^2+B^2−2AB`
• `(A+B)(A−B)=A^2−B^2`

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Algebra 1a: Chapter 7, Lesson 2, page 309.

Graphing Equations

We can graph an equation, by building a T-chart of values for both x and y. You can choose any values for `x ` and `y` when making your T-chart. I like to use values like 0, 1, and 2. Make them easy and try to pick AT LEAST 3 points when graphing and equation. YOU MUST USE A RULER WHEN CONNECTING THE DOTS TOO!

Sometimes, it can be easier when building the T-chart to “solve for `y`” first, this just cuts down on the workload. Solving for `y` means isolating the `y` variable to one side of the equation and keeping the constants and ALL other variables on the other side.

Here are 2 good links from purplemath.com, the first about graphing in general (lesson 1) and the second about the T-charts and lesson 2!

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

Algebra: Chapter 5 Review

Multiplying Polynomials

To multiply a monomial and a polynomial: multiply each term of the polynomial by the monomial

There are 3 techniques to multiply binomials:

1. FOIL (FIRST, OUTSIDE, INSIDE, LAST – or its derivatives for trinomials)
   • Multiply each term of a polynomial by EVERY OTHER TERM of the other polynomial
2. The BOX method
   • The BOX method, more like a rectangle with each term representing 1 side of an inner box. A binomial multiplied with a trinomial will be a BOX containing 2 x 3 number of smaller boxes inside it. Each term represents 1 edge in distance in the inner boxes.
3. The old fashioned multiplication method outlined on page 249.

You get to chose which is most comfortable for you BUT REMEMBER THE BOX for Chapter 6!

Remember too, the shortcuts for special binomials:

• `(A+B)(A+B)=(A+B)^2=A^2+B^2+2AB`
• `(A−B)(A−B)=(A−B)2=A^2+B^2−2AB`
• `(A+B)(A−B)=A^2−B^2`

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Algebra 1a: Chapter 7, Lesson 1, page 304

Coordinates

Coordinates are defined as `(x, y)` where the x-axis runs left to right and the y-axis runs up and down. The origin is where the points `(0, 0)` exists. A fancy word for the x-axis is the abscissa and the y-axis is the ordinate. There are 4 quadrants:

• I – both x and y axis have positive value (upper right)
• II- x axis is negative and y axis is positive (upper left)
• III – both x and y axis have negative values (lower left)
• IV – x axis has positive value, while the y axis has negative value (lower right)

Algebra: Chapter 5, Lesson 11, page 249 (day 2).

Multiplying Polynomials

To multiply a monomial and a polynomial: multiply each term of the polynomial by the monomial

There are 3 techniques to multiply binomials:

1. FOIL (FIRST, OUTSIDE, INSIDE, LAST – or its derivatives for trinomials)
   • Multiply each term of a polynomial by EVERY OTHER TERM of the other polynomial
2. The BOX method
   • The BOX method, more like a rectangle with each term representing 1 side of an inner box. A binomial multiplied with a trinomial will be a BOX containing 2 x 3 number of smaller boxes inside it. Each term represents 1 edge in distance in the inner boxes.
3. The old fashioned multiplication method outlined on page 249.

You get to chose which is most comfortable for you BUT REMEMBER THE BOX for Chapter 6!

Remember too, the shortcuts for special binomials:

• `(A+B)(A+B)=(A+B)^2=A^2+B^2+2AB`
• `(A−B)(A−B)=(A−B)2=A^2+B^2−2AB`
• `(A+B)(A−B)=A^2−B^2`

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

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Algebra 1a: Chapter 4 –  200 POINT BENCHMARK!

Algebra: Chapter 5, Lesson 11, page 249.

Multiplying Polynomials

To multiply a monomial and a polynomial: multiply each term of the polynomial by the monomial

There are 3 techniques to multiply binomials:

1. FOIL (FIRST, OUTSIDE, INSIDE, LAST – or its derivatives for trinomials)
   • Multiply each term of a polynomial by EVERY OTHER TERM of the other polynomial
2. The BOX method
   • The BOX method, more like a rectangle with each term representing 1 side of an inner box. A binomial multiplied with a trinomial will be a BOX containing 2 x 3 number of smaller boxes inside it. Each term represents 1 edge in distance in the inner boxes.
3. The old fashioned multiplication method outlined on page 249.

You get to chose which is most comfortable for you BUT REMEMBER THE BOX for Chapter 6!

Remember too, the shortcuts for special binomials:

• `(A+B)(A+B)=(A+B)^2=A^2+B^2+2AB`
• `(A−B)(A−B)=(A−B)2=A^2+B^2−2AB`
• `(A+B)(A−B)=A^2−B^2`

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

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Algebra 1a: Chapter 4, Review for TOMORROW’s 200 POINT BENCHMARK!

Chapter Review – 2 Step Equations & Inequalities

Follow these steps and you can’t go wrong … take your time!

1. If you have an absolute value, `|x|` for the variable, remove it (ignore it) and rewrite as if it didn’t exist.
2. Multiply both sides to clear fractions or decimals, if necessary. Multiply by the LCD for fractions or by `10` or `100` for decimals.
3. Distribute (whacky arrows) if you see parenthesis `( )` or square brackets `[ ]`
4. Collect like terms on each side, if necessary.
5. Use the addition or subtraction property to move the variable to one side and all other terms and constants to the other side of the equation.
6. Use the multiplication or division or reciprocal (good for fractions) properties to solve for the variable.
7. ONLY FOR INEQUALITIES – if you multiply or divide by a negative number in step 6, DON’T FORGET TO FLIP THE SIGN OF THE INEQUALITY!
8. If this was an absolute value problem `|x|`, re-insert the absolute value and check your answers, there may be 2 of them (`±`).

Word Problem Tips

Addition: plus, sum, more than, increased by, total, in all

Subtraction: minus, difference, less than, subtract, decreased by

Multiplication: times, product, multiplied, each, of

Division: dividend, quotient

Algebra: Chapter 5, Lesson 10, page 245.

Multiplying Binomials: Special Products

We can use the FOIL or BOX methods to multiply binomials. There are however, 3 special cases that if you recognize them, will save you some work:

1. `(A+B)(A-B)=A^2−B^2`
2. `(A+B)(A+B)=A^2+B^2+2AB`
3. `(A−B)(A−B)=A^2+B^2−2AB`

Remember, that `A` and `B` are the terms (monomials or constants) that we just plug in. If you forget these 3 cases, that’s OK, just FOIL or BOX the binomials and the answers are still the same.

For example:

`(2x + 7)(2x – 7)` where `2x = a` and `7 = b`, we have an answer of `4x^2 – 49`.

Another example:

`(2x + 7)(2x +7)` where again `2x = a` and `7 = b`, we have an answer of `4x^2 + 49 + 2*2x*7` or `4x^2 +28x + 49`!

Go slow and you’ll be fine.

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

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Algebra 1a: Chapter 4, Lesson 6, page 194.

Reasoning Strategies

Three steps to follow:

1. UNDERSTAND the problem
2. Develop and carry out a PLAN
3. Find the ANSWER and CHECK

You can also these strategies:

• Draw a diagram
• Make an organized list
• Look for a pattern
• Try, test and revise
• Use logical reasoning
• Simplify the problem
• Write an equation
• Make a table
• Work backward!

Algebra: Chapter 5, Lesson 9, page 240.

Multiplication of Monomials and Binomials

If we multiply a monomial and a binomial, all we have to do is DISTRIBUTE the monomial through the binomial, pretty simple to do (REMEMBER: the CRAZY arrows – too much coffee!).

FOIL – first, outside, inside, last. A handy way to multiply 2 binomials. If the 2 binomials are:

`(A + B) * (C + D)`, then FOIL gives us `AC + AD + BC + BD` where A, B, C and D represent the terms in each binomial.

We can also do the BOX METHOD which is explained on page 240 at the bottom.  I talked about it today. It’s basically representing the binomials as the dimensions of a box which itself contains 4 smaller boxes as defined by the 4 terms. The BOX METHOD finds the area of each of the 4 boxes and then the addition of all 4, gives us the area of the 2 binomials multiplied together. Its hard to describe here.

Truthfully, if you get in the habit of using the BOX METHOD, then FACTORING in Chapter 6 will be a whole lot easier!

Monomials multiplied with binomials are described here. Binomials using FOIL are described with more examples here!

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

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Algebra 1a: Chapter 4, Lesson 6, page 194.

Reasoning Strategies

Three steps to follow:

1. UNDERSTAND the problem
2. Develop and carry out a PLAN
3. Find the ANSWER and CHECK

You can also these strategies:

• Draw a diagram
• Make an organized list
• Look for a pattern
• Try, test and revise
• Use logical reasoning
• Simplify the problem
• Write an equation
• Make a table
• Work backward!

Algebra: Chapter 5, Lesson 8, page 236.

Subtraction of Polynomials

We do just like the addition of polynomials, except for subtraction, we (remember way back to subtracting with negative numbers) add the opposite. The other thing we have to remember is the sign problems for double negatives, make “Change-change” or “bling-bling” with 2 negatives.

You can do the subtraction horizontally or vertically. Horizontally requires that you scan across the polynomials. It’s easier for me to put the polynomials in descending order and then combine like terms, remembering the combining terms MUST HAVE THE SAME VARIABLE AND THE SAME EXPONENTS!

If you do the problems horizontally, remember that the `-` sign turns everything in 2nd polynomial’s parenthesis `( x^2 …)` to its opposite sign!

Vertically (or columns in the textbook) require you to write out the problem with spaces for missing terms. It is easier to line up terms this way and you can stick in other terms as well that are not common to both polynomials.

See here for 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 5, page 187.

Using Inequalities

We learned key phrases for lesson 5 (word translation problems):

• “Less than or equal to”, “is at most”, “no more than” — ≤
• “No less than”, “at least”, “more than or equal to” — ≥
• “Is less than” — <
• “Is greater than” — >

We learned to read the problem, draw a picture or understand what is being asked of us before we start solving an equation or inequality.

Remember for 2 step inequalities, 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.

Here are some keyword descriptions from purplemath.com to help us with word problems (ugh …)

Two of tonight’s Lesson 5 homework problems solved by MrE are here! Just click it!

Algebra: Chapter 5, Lesson 7, page 231.

Addition of Polynomials

When we add polynomials, it can be easier if we order them in descending order for one of the variables. We can add polynomials in 2 ways:

1. Horizontally – scan across the polynomials and see what terms can be combined. REMEMBER, we can add terms ONLY IF they have the SAME VARIABLES and the SAME EXPONENTS.
2. Vertically – leave space for terms that are missing when making the columns, see the examples in the textbook on pagee 232.

Here is a link with more examples.

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

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Algebra 1a: Chapter 4,  Lesson 5, page 187.

Using Inequalities

We learned key phrases for lesson 5 (word translation problems):

• “Less than or equal to”, “is at most”, “no more than” — ≤
• “No less than”, “at least”, “more than or equal to” — ≥
• “Is less than” — <
• “Is greater than” — >

We learned to read the problem, draw a picture or understand what is being asked of us before we start solving an equation or inequality.

Remember for 2 step inequalities, 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.

Here are some keyword descriptions from purplemath.com to help us with word problems (ugh …)

Two of tonight’s Lesson 5 homework problems solved by MrE are here! Just click it!

Algebra: Chapter 5, Lesson 6, page 226.

We can write polynomials is ascending or descending order. For descending order, the term with the greatest exponent for our variable of interest is first, the term with the next greatest exponent for x is second and ….

`2x^4 − 12x^3 + (1/2)x^2 − 2x + 14` for example!

Finally, we can evaluate polynomials when we replace the variable by a number and calculate the resulting answer. This is called EVALUATING THE POLYNOMIAL.

Click here for some examples and hints from purplemath.com

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!