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:
- 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.
- 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:
- FOIL (FIRST, OUTSIDE, INSIDE, LAST – or its derivatives for trinomials)
- Multiply each term of a polynomial by EVERY OTHER TERM of the other polynomial
- 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.
- 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