MrE's Algebra Blog

Algebra: Chapter 6, Lesson 8, page 286.

Solving Equations by Factoring

The principle of zero products states that: for any rational numbers a and b, if the product `ab = 0`, then `a=0` or `b=0` or both `a` and `b = 0`.

Factor and solve equations by using the following:

1. Get `0` on one side of the equation using the addition property
2. FACTOR the expression on the other side of the equation
3. Set each factor equal to zero
4. Solve each equation.

If we have an equation with `0` on one side and a factorization on the other side, we can solve the equation by finding the values that make the factors `0`. This is even easier!

Since we are solving quadratic equations (they are NOT linear because they have an exponent that is squared, `x^2`), we can have at MOST 2 solutions. We can have NO solutions, 1 solution or 2 solutions.

Remember to use all the techniques you know to factor!

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

======================================================================

Algebra 1a: Chapter 7, Lesson 7, page 333.

Fitting Equations to Data

“The mathematical relationship between 2 variables is of interest in many real-world situations. The relationship between 2 variables can often be expressed as a linear equation, which is calleda model of the situation. The model can be use to make estimates or predictions about the quantities represented by the variables.”

In the problems with real-world data, we sometimes have to approximate, by plotting the data on a x-y graph and then drawing (fitting) a line the best way we can through MOST of the data. We can then use 2 of the points on our line to use to develop our linear equation.

We can use either the slope-intercept equation (`y = mx + b`) or the point-slope equation [`y – y_1 = m(x – x_1)`] to develop our linear equation.

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

Algebra: Chapter 6, Lesson 7, page 283.

Factoring: A General  Strategy

To factor polynomials:

• Always look first for a common factor
• Then look at the number of terms
  • 2 terms – determine whether you have a difference of 2 squares
  • 3 terms – determine whether the trinomial is a square of a binomial. If not, test the factors of the terms.
• Always factor completely!

Use all the strategies we’ve learned so far to factor a variety of problems. Don’t forget to use:

• Monomial factorization (lesson 6-1)
• The differences of 2 squares (lesson 6-2), `(a^2 – b^2) = (a – b)(a + b)`
• Trinomial squares (lesson 6-3), `a^2 + 2ab + b^2 = (a + b)^2` or with a negative `(-2ab)`
• The BOX METHOD (lesson 6-4 and 6-5) for `x^2 + bx + c` or `ax^2 + bx + c` type of equations
• Factoring by grouping (lesson 6-6) for polynomials with 4 or more terms.

The toughest part is figuring out what technique to use! Go slow and you’ll be OK!

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

======================================================================

Algebra 1a: Chapter 7, Lesson 7, page 333.

Fitting Equations to Data

“The mathematical relationship between 2 variables is of interest in many real-world situations. The relationship between 2 variables can often be expressed as a linear equation, which is calleda model of the situation. The model can be use to make estimates or predictions about the quantities represented by the variables.”

In the problems with real-world data, we sometimes have to approximate, by plotting the data on a x-y graph and then drawing (fitting) a line the best way we can through MOST of the data. We can then use 2 of the points on our line to use to develop our linear equation.

We can use either the slope-intercept equation (`y = mx + b`) or the point-slope equation [`y – y_1 = m(x – x_1)`] to develop our linear equation.

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

Algebra: Chapter 6, Lesson 6, page 281.

Factoring by Grouping

Factoring by grouping works for polynomials that are greater than trinomials. Grouping usually works with polynomials that have 4 terms. All that you have to do is group the polynomials into binomials using parentheses. But, remember, that not all 4 term expressions can be factored this way.

Example: `6x^3+9x^2+4x+6` becomes `(6x^3+9x^2)+(4x+6)`.

We can factor out a `3x^2` from the first binomial and a `2` from the second binomial leaving us with `(3x^2)(2x+3)+2(2x+3)` or factoring out the `(2x+3)`, we have`(2x+3)(3x^2+2)`.

Here is a link from purplemath.com. She calls it factoring in “pairs”.

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

======================================================================

Algebra 1a: Chapter 7, Lesson 6, page 328, Day 1 – 3.

Finding an Equation of a Line

There are 2 ways that we can find an equation of a line. However, we need to have at least 2 points `(x,y)` or 1 point `(x,y)` and the slope, `m` provided. If we have those things, we can find the equation by the 2 methods below, you choose the one you like:

A. Using the slope-intercept equation, `y=mx+b`

1. (Don’t forget, for this to work, you have to be given the slope, `m`, and at least 1 `(x,y)` point.)
2. If you have the slope, `m`, plug it in for `m` above and pick the `(x,y)` that correspond to the point given.
3. In the equation then, you have the `y`, `m` and `x` known.
4. All you have to do is solve for `b`, the y-intercept.
5. Solve for `b`, then plug in the `b` and `m` into the slope-intercept equation.

WARNING: IF you are not given the slope, then you are given 2 points. Given the 2 points, find the slope `m` with the equation `m=(y_2−y_1)/(x_2−x_1)`, then proceed as in step 2 above.

With this method, you have to solve for b, the y-intercept.

B. Using the point-slope equation (which is a derivation of the slope definition), `(y−y_1)=m(x−x_1)`

1. (Don’t forget, for this to work, you need the slope and 1 point or at least 2 points from which you can find the slope.)
2. (Notice too, that there is NO LONGER a `y_2` and `x_2`, just a `y` and `x`. LEAVE IT THAT WAY!)
3. If you have the slope `m`, use it. If you have 2 points, then find the slope – like the WARNING above.
4. Choose 1 of the `(x, y)` points to use for `(x_1, y_1)` and plug in the values that you know (`x_1`, `y_1`, and `m`).
5. Solve the equation for `y` and remember that you have to distribute on the right side!

With this method you have to you the distribution method on the right. You DO NOT find the `b` or y-intercept.

Either method works, you choose what is most comfortable for YOU!

Here is a link to both methods from purplemath.com.

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

Algebra: Chapter 6, Lesson 5, page 278, Day 1 – 2.

Factoring `ax^2 + bx + c`

Go slow, be methodical and do NO shortcuts. The Box and the Cross are your friends when we have `ax^2` instead of just `x^2`.

If we have a trinomial like `6m^2 + 15mn − n^2`, then the factors in the middle term (of the cross) will be like `mn` instead of just `x`!

Here is a link to my podcast on iTunes about the BOX and CROSS METHODS.

======================================================================

Algebra 1a: Chapter 7, Lesson 6, page 328, Day 1 – 3.

Finding an Equation of a Line

There are 2 ways that we can find an equation of a line. However, we need to have at least 2 points `(x,y)` or 1 point `(x,y)` and the slope, `m` provided. If we have those things, we can find the equation by the 2 methods below, you choose the one you like:

A. Using the slope-intercept equation, `y=mx+b`

1. (Don’t forget, for this to work, you have to be given the slope, `m`, and at least 1 `(x,y)` point.)
2. If you have the slope, `m`, plug it in for `m` above and pick the `(x,y)` that correspond to the point given.
3. In the equation then, you have the `y`, `m` and `x` known.
4. All you have to do is solve for `b`, the y-intercept.
5. Solve for `b`, then plug in the `b` and `m` into the slope-intercept equation.

WARNING: IF you are not given the slope, then you are given 2 points. Given the 2 points, find the slope `m` with the equation `m=(y_2−y_1)/(x_2−x_1)`, then proceed as in step 2 above.

With this method, you have to solve for b, the y-intercept.

B. Using the point-slope equation (which is a derivation of the slope definition), `(y−y_1)=m(x−x_1)`

1. (Don’t forget, for this to work, you need the slope and 1 point or at least 2 points from which you can find the slope.)
2. (Notice too, that there is NO LONGER a `y_2` and `x_2`, just a `y` and `x`. LEAVE IT THAT WAY!)
3. If you have the slope `m`, use it. If you have 2 points, then find the slope – like the WARNING above.
4. Choose 1 of the `(x, y)` points to use for `(x_1, y_1)` and plug in the values that you know (`x_1`, `y_1`, and `m`).
5. Solve the equation for `y` and remember that you have to distribute on the right side!

With this method you have to you the distribution method on the right. You DO NOT find the `b` or y-intercept.

Either method works, you choose what is most comfortable for YOU!

Here is a link to both methods from purplemath.com.

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

Algebra: Chapter 6, Lesson 4, page 273 (day #3).

Factoring quadratics of the form … `x^2 + bx +c `

The BOX METHOD is linked here, all you need is a computer with sound.

You can also grab it for your iPod/iPod Touch/iPhone on iTunes if you search “MrE Algebra” on the iTunes store!!

======================================================================

Algebra 1a: Chapter 7, Lesson 5, page 323.

Equations and Slope

An equation `y=mx+b` is called the slope-intercept equation of a line. The slope is `m` and the y-intercept is `b`. Without having to plot points, or make a T chart, we can easily determine the slope as the coefficient in front of the `x` variable and the y-intercept as `(0, b)`, the constant in the slope-intercept equation.

If the equation is not of the slope-intercept form, solve for `y` to isolate it, just like we have done in the past. The key is to have the `y` on one side of the equation and the `x` and its coefficient and the constant `b` on the other side. Usually, you have to add/subtract terms first, then multiply/divide by the coefficient in front of the `y`.

You can easily plot an equation. Start with the `(0, b)` y-intercept and then use the slope definition of `m=(rise)/(run)` to move up/down and then left/right on the graph paper as determined by the values of the rise and run. Remember to watch the signs of the rise and run.

For example, find the slope of:

`2x + 3y = 7`

first subtract `2x` from both sides

`2x – 2x + 3y = -2x +7`

to give us

`3y = -2x + 7`

divide both sides by `3` to isolate the `y`

`y = (-2/3)x + 7/3`

and the slope is then `(-2/3)` and the y-intercept is `(0, 7/3)`

See these examples from purplemath.com too! Here are others to help you graph equations given the slope m and the y-intercept b.

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

Algebra: Chapter 6, Lesson 4, page 273 (day #2).

Factoring quadratics of the form … `x^2 + bx +c `

The BOX METHOD is linked here, all you need is a computer with sound.

You can also grab it for your iPod/iPod Touch/iPhone on iTunes if you search “MrE Algebra” on the iTunes store!!

======================================================================

Algebra 1a: Chapter 7, Lesson 5, page 323.

Equations and Slope

An equation `y=mx+b` is called the slope-intercept equation of a line. The slope is `m` and the y-intercept is `b`. Without having to plot points, or make a T chart, we can easily determine the slope as the coefficient in front of the `x` variable and the y-intercept as `(0, b)`, the constant in the slope-intercept equation.

If the equation is not of the slope-intercept form, solve for `y` to isolate it, just like we have done in the past. The key is to have the `y` on one side of the equation and the `x` and its coefficient and the constant `b` on the other side. Usually, you have to add/subtract terms first, then multiply/divide by the coefficient in front of the `y`.

You can easily plot an equation. Start with the `(0, b)` y-intercept and then use the slope definition of `m=(rise)/(run)` to move up/down and then left/right on the graph paper as determined by the values of the rise and run. Remember to watch the signs of the rise and run.

For example, find the slope of:

`2x + 3y = 7`

first subtract `2x` from both sides

`2x – 2x + 3y = -2x +7`

to give us

`3y = -2x + 7`

divide both sides by `3` to isolate the `y`

`y = (-2/3)x + 7/3`

and the slope is then `(-2/3)` and the y-intercept is `(0, 7/3)`

See these examples from purplemath.com too! Here are others to help you graph equations given the slope m and the y-intercept b.

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

Algebra: Chapter 6, Lesson 4, page 273.

Factoring quadratics of the form … `x^2 + bx +c `

The BOX METHOD is linked here, all you need is a computer with sound.

You can also grab it for your iPod on iTunes if you search “MrE Algebra” on the iTunes store!!

======================================================================

Algebra 1a: Chapter 7, Lesson 4, page 318.

Slope of a Line

The slope of a line is the tilt or slant. The slope is defined as the letter `m` and `m=(rise)/(run)`, where the `rise` is the change in the y-coordinate and the `run` is the change in the x-coordinate. This definition works great when you have 2 points on graph paper. You can just count the squares for the rise and the run. It doesn’t matter too, which point you start with.

Remember, slope is positive if the line is going from lower left to upper right. The slope is negative if its coming from upper left to lower right.

A slope of  `0` is a horizontal line and a line with NO SLOPE is a vertical line.

Another definition of slope, given 2 point and NO GRAPH PAPER is:

`m=(y_2−y_1)/(x_2−x_1)`, where `(x_1,y_1)` and `(x_2,y_2)` are 2 points on the line.

For example, for the points (2, -3) and (-4, 2), the slope is:

`m = (2 – (-3))/(-4 – 2) = 5/(-6) = -(5/6)`

Purplemath has this link as well.

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

Algebra: Chapter 6, Lesson 3, page 270

Trinomial Squares

We are going backward again, using the properties:

• `(a + b)(a + b) = a^2 + 2ab + b^2 = (a + b)^2`
• `(a – b)(a – b) = a^2 – 2ab + b^2 = (a – b)^2`

For a trinomial square to factor, we must make sure that:

• 2 of the terms must be squares, `a^2` and `b^2`
• There must be NO MINUS sign before the `a^2` and `b^2`
• If we multiply `a` and `b` and double the result, we get the 3rd term, `2ab` or its additive inverse `- 2ab`.

Sometimes, we can also factor out a coefficient in front of the `a^2`, like `2a^`. We MIGHT be able to factor out the `2` before we start the trinomial determination.

Here is a purplemath link that describes trinomial squares, scroll down about 1/2 way to get to the information.

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

======================================================================

Algebra 1a: Chapter 7, Lesson 4, page 318.

Slope of a Line

The slope of a line is the tilt or slant. The slope is defined as the letter `m` and `m=(rise)/(run)`, where the `rise` is the change in the y-coordinate and the `run` is the change in the x-coordinate. This definition works great when you have 2 points on graph paper. You can just count the squares for the rise and the run. It doesn’t matter too, which point you start with.

Remember, slope is positive if the line is going from lower left to upper right. The slope is negative if its coming from upper left to lower right.

A slope of  `0` is a horizontal line and a line with NO SLOPE is a vertical line.

Another definition of slope, given 2 point and NO GRAPH PAPER is:

`m=(y_2−y_1)/(x_2−x_1)`, where `(x_1,y_1)` and `(x_2,y_2)` are 2 points on the line.

For example, for the points (2, -3) and (-4, 2), the slope is:

`m = (2 – (-3))/(-4 – 2) = 5/(-6) = -(5/6)`

Purplemath has this link as well.

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

Algebra: Chapter 6, Lesson 2, page 266.

Difference of 2 squares

For a binomial to be a difference of 2 squares, 2 conditions must be met.

• There must be 2 terms, both must be square (e.g., `4x^2` and `9x^4`)
  There must be a minus sign `(-)` between the 2 terms.

We are going backward in our factoring, using our first Chapter 5 shortcut formula:

`a^2 – b^2 = (a + b)(a – b)`,

so once we have the 2 squares, just plug them in to our formula!

Examples:

`9a^8b^4 – 49 = (3a^4b^2)^2 – 7^2`

`= (3a^4b^2 + 7)(3a^4b^2 – 7)`

and another example where we have to factor something out first, namely `x^4`, we have:

`49x^4 – 9x^6 = x^4(49 – 9x^2) `

`= x^4[7^2 – (3x)^2]`

with the final factors being = `x^4(7 +3x)(7 – 3x)`

More info is here too!

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

======================================================================

Algebra 1a: Chapter 7, Lesson 3, page 313.

Linear Equations and their Graphs

Linear equation have to have variables with a power of 1, NO mixed variable products and NO variables in an equation in the denominator. The easiest way to plot or graph an equation is to use the x-intercept and y-intercept.

• The x-intercept of a line is the x-coordinate of the point where the line intercepts the x-axis. To do this, all we have to do is set `y=0` and solve for `x`.
• The y-intercept of a line is the y-coordinate of the point where the line intercepts the y-axis. To do this, set `x=0` and solve for `y`

The standard form of a linear equation in 2 variables is `Ax + By = C`, where A, B and C are constants.

For horizontal lines, the graph of `y = b` is the x-axis or a line parallel to the x-axis with y-intercept, `b`.

For vertical lines. the graph of `x = a` is the y-axis or a line parallel to the y-axis with x-intercept, `a`.

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

Algebra: Chapter 6, Lesson 1, page 262.

Factoring Polynomials

Divisibility Rules

A number is divisible by the numbers below if the following rules hold:

• 2, if the last digit is an even number
• 3, if the sum of the digits is divisible by 3
• 4, if the number formed by the last 2 digits is divisible by 4
• 5, if the number ends in 0 or 5
• 7, the Nike Rule “just do it”, the long division that is
• 9, if the sum of the digits is divisible by 9, similar to the 3 rule above
• 10, if the last digit is 0.

Factoring is the reverse of multiplying. To factor an expression mean to write an equivalent expression that is the product of 2 or more expressions.

To factor a monomial, we find 2 monomials whose product is that monomial. For example `20x^2` has as factors `(4x)(5x)` or `(2x)(10x)` or `(x)(20x)`.

Remember, to multiply a monomial and a polynomial, we use the distributive property to multiply each term of the polynomial by the monomial.

To FACTOR, we do the reverse and FACTOR OUT a common factor. We use the factor COMMON to EACH TERM with the greatest possible coefficient and the variable to the GREATEST POWER.

For example, `16a^2b^2 + 20a^2`

We can re-write it as

`4*4*a^2b^2 + 4*5*a^2`.

The terms that are common are `4a^2` because they are in both terms. So … we can re-write it again (taking out the `4a^2` and putting it on the outside of the parenthesis as

`4a^2(4b^2 + 5)` and that is our FACTORED ANSWER!

Factoring is also described here.

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

======================================================================

Algebra 1a: Chapter 7, Lesson 3, page 313.

Linear Equations and their Graphs

Linear equation have to have variables with a power of 1, NO mixed variable products and NO variables in an equation in the denominator. The easiest way to plot or graph an equation is to use the x-intercept and y-intercept.

• The x-intercept of a line is the x-coordinate of the point where the line intercepts the x-axis. To do this, all we have to do is set `y=0` and solve for `x`.
• The y-intercept of a line is the y-coordinate of the point where the line intercepts the y-axis. To do this, set `x=0` and solve for `y`

The standard form of a linear equation in 2 variables is `Ax + By = C`, where A, B and C are constants.

For horizontal lines, the graph of `y = b` is the x-axis or a line parallel to the x-axis with y-intercept, `b`.

For vertical lines. the graph of `x = a` is the y-axis or a line parallel to the y-axis with x-intercept, `a`.

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