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As we delve deeper into various ways to express a mathematical quantity, we will see that many of the basic math properties we have learned in recent years are still the foundation for working with polynomials. Essential Understanding: Polynomials are classified by the number of terms, or monomials they are comprised of. 1st degree: (when the exponent is 1), 2nd degree (when the exponent is 2), 3rd degree (when, well you get the idea). Where it becomes confusing: The degree of a polynomial is determined from the degree of the monomial contained within it that is highest. Essential Understanding: The distributive property can be utilized to multiply a monomial by a polynomial.

Essential Understanding: When you factor a polynomial you are reversing the process to identify the individual factors comprising the polynomial. Consider the product we arrived at above: 12x2 + 9x What factors are common to both terms? It doesna€™t matter whether you are factoring a binomial or a polynomial with 20 terms (only it will take more time and brain cells), the concept is the same. Essential Understanding: There are several ways to find the products of polynomials, and circumstances can make some approaches more efficient.

Challenge: Multiply 59 x 61 without a calculator or use of traditional multiplication methods. Now if we think of the apples potentially being two separate things, we could also think of them as a binomial.

Example: (3x+2) (2x-1) Think of the 3x as the 4 and the 2 as the 5, with the 2x-1 being the apples. There is another essential method which everyone must master in order to successfully be able to factor polynomials later. This method is referred to as FOIL which stands for First, Outer, Inner and Last or First, Outside, Inside and Last which is the same idea. This results in: 6x2 - 3x + 4x - 2 and after you combine like terms you end up with the same result as before. Now consider multiplying two polynomials: ( 2x + 2 ) ( x2 + 3x - 4 ) Actually I am going to use a table to do this multiplication, mostly because it would probably end up all jumbled on various web browsers if I didna€™t. Essential Understanding: There are special rules you can use to simplify the square of a binomial or the product of a sum and a difference. Therefore you can conclude that if you are a€?Squaring a sum,a€? you square the first term, double the product of the first and last terms, and square the last terms. There is only one difference (no pun intended) when a€?Squaring a difference.a€? The middle term of the resulting trinomial ends up negative.

Essential Understanding: Some trinomials written in this form may be factored as two binomials.

So now that youa€™ve had a chance to enjoy your favorite video teachers, here is my a€?thinkinga€? approach.

First consider the first term of the trinomial x2, there are only two possible First terms for the binomials that would product that product: x a€? x, so our binomials start off looking like this ( x ) ( x ). You might forget the rules in the text, but you can remember this a€?thinkinga€? process forever. Essential Understanding: Trinomials written in this form have a coefficient other than one for the first (quadratic) term.

Once again the book will present this as different approaches to different scenarios and the homework videos present yet another approach, but I prefer to address it as a thinking process that works for all scenarios. The first thing you might have observed is that if you tried to foil these, since there is a coefficient for the first term, it dramatically increases the number of combinations of factors. So now that youa€™ve had a chance to enjoy your favorite video teachers again, here is my recommended approach. Essential Understanding: Some trinomials can be factored by reversing the process by which you multiplied those special cases previously. A Perfect Square Trinomial was the resulting product when you squared a sum or a difference in lesson 8.4. Essential Understanding:A A Polynomials are classified by the number of terms, or monomials they are comprised of. Essential Understanding:A A The distributive property can be utilized to multiply a monomial by a polynomial.

Essential Understanding:A A When you factor a polynomial you are reversing the process to identify the individual factors comprising the polynomial.

Essential Understanding:A A There are several ways to find the products of polynomials, and circumstances can make some approaches more efficient. Essential Understanding:A A There are special rules you can use to simplify the square of a binomial or the product of a sum and a difference.

Essential Understanding:A A Some trinomials written in this form may be factored as two binomials. Essential Understanding:A A Trinomials written in this form have a coefficient other than one for the first (quadratic) term. Essential Understanding:A A Some trinomials can be factored by reversing the process by which you multiplied those special cases previously.

In this case the first one is third degree, the second is second degree and the last is first degree, therefore the polynomial is a third degree polynomial. Only variable terms with the same variable raised to the same power can be added or subtracted.

If you break the first binomial up into its parts, you can use the distributive property to multiply the first by the second. This can be used to multiply any polynomials, but it gets pretty messy the more terms there are in the polynomials. Later on we will discover how rethinking this process in reverse can help us to factor a binomial.

Your text suggests different approaches given different scenarios regarding the signs of the terms in the trinomial you are trying to factor. If it is negative, and since you are multiplying the Last terms of the binomials to arrive at it, one of the last terms needs to be positive and one negative: ( x + ) ( x - ).

Keeping in mind that these two values when added produce the a€?ba€? value of the middle term. There are only two ways this can happen, Since it is the product of the Last terms of the binomials, they are either both positive or both negative. Books will usually have you looking for factors of the second binomial, but the reality is that if this trinomial will factor, the second binomial must match the first (x + 5). Is it double the square root of the first term (x) times the square root of the second term (3)? If you have not figured it out yet, this process will only work when multiplying two binomials.

Therefore, if the a€?ba€? of the middle term is positive, the larger number must go into the first binomial.

Even in math new things come along once in a great while, and factoring these trinomials is as cool as sliced bread. Sure you could multiply 30 times (- 8), but our objective is always to make life easy for ourselves.

You should have determined that it is a quadratic binomial since it is of the second degree and has two terms. Write 3x, then open a parenthesis and determine what 3x must be multiplied by to achieve the polynomial you started with. All you need to do is to recognize these special cases and you can let your brain do all the work instead of using up pencil lead. If the a€?ba€? of the middle term is negative, the larger number must go into the second binomial.

You should notice that 2 is a common factor to all three terms, therefore you should factor it out first. Since they disappeared you could conclude that you ended up with the square of the first term less the square of the second term: 3600 - 1.

Therefore whatever the sign of the middle term, the last terms of the binomial factors will be the same.

Essential Understanding: When you factor a polynomial you are reversing the process to identify the individual factors comprising the polynomial. Consider the product we arrived at above: 12x2 + 9x What factors are common to both terms? It doesna€™t matter whether you are factoring a binomial or a polynomial with 20 terms (only it will take more time and brain cells), the concept is the same. Essential Understanding: There are several ways to find the products of polynomials, and circumstances can make some approaches more efficient.

Challenge: Multiply 59 x 61 without a calculator or use of traditional multiplication methods. Now if we think of the apples potentially being two separate things, we could also think of them as a binomial.

Example: (3x+2) (2x-1) Think of the 3x as the 4 and the 2 as the 5, with the 2x-1 being the apples. There is another essential method which everyone must master in order to successfully be able to factor polynomials later. This method is referred to as FOIL which stands for First, Outer, Inner and Last or First, Outside, Inside and Last which is the same idea. This results in: 6x2 - 3x + 4x - 2 and after you combine like terms you end up with the same result as before. Now consider multiplying two polynomials: ( 2x + 2 ) ( x2 + 3x - 4 ) Actually I am going to use a table to do this multiplication, mostly because it would probably end up all jumbled on various web browsers if I didna€™t. Essential Understanding: There are special rules you can use to simplify the square of a binomial or the product of a sum and a difference. Therefore you can conclude that if you are a€?Squaring a sum,a€? you square the first term, double the product of the first and last terms, and square the last terms. There is only one difference (no pun intended) when a€?Squaring a difference.a€? The middle term of the resulting trinomial ends up negative.

Essential Understanding: Some trinomials written in this form may be factored as two binomials.

So now that youa€™ve had a chance to enjoy your favorite video teachers, here is my a€?thinkinga€? approach.

First consider the first term of the trinomial x2, there are only two possible First terms for the binomials that would product that product: x a€? x, so our binomials start off looking like this ( x ) ( x ). You might forget the rules in the text, but you can remember this a€?thinkinga€? process forever. Essential Understanding: Trinomials written in this form have a coefficient other than one for the first (quadratic) term.

Once again the book will present this as different approaches to different scenarios and the homework videos present yet another approach, but I prefer to address it as a thinking process that works for all scenarios. The first thing you might have observed is that if you tried to foil these, since there is a coefficient for the first term, it dramatically increases the number of combinations of factors. So now that youa€™ve had a chance to enjoy your favorite video teachers again, here is my recommended approach. Essential Understanding: Some trinomials can be factored by reversing the process by which you multiplied those special cases previously. A Perfect Square Trinomial was the resulting product when you squared a sum or a difference in lesson 8.4. Essential Understanding:A A Polynomials are classified by the number of terms, or monomials they are comprised of. Essential Understanding:A A The distributive property can be utilized to multiply a monomial by a polynomial.

Essential Understanding:A A When you factor a polynomial you are reversing the process to identify the individual factors comprising the polynomial.

Essential Understanding:A A There are several ways to find the products of polynomials, and circumstances can make some approaches more efficient. Essential Understanding:A A There are special rules you can use to simplify the square of a binomial or the product of a sum and a difference.

Essential Understanding:A A Some trinomials written in this form may be factored as two binomials. Essential Understanding:A A Trinomials written in this form have a coefficient other than one for the first (quadratic) term. Essential Understanding:A A Some trinomials can be factored by reversing the process by which you multiplied those special cases previously.

In this case the first one is third degree, the second is second degree and the last is first degree, therefore the polynomial is a third degree polynomial. Only variable terms with the same variable raised to the same power can be added or subtracted.

If you break the first binomial up into its parts, you can use the distributive property to multiply the first by the second. This can be used to multiply any polynomials, but it gets pretty messy the more terms there are in the polynomials. Later on we will discover how rethinking this process in reverse can help us to factor a binomial.

Your text suggests different approaches given different scenarios regarding the signs of the terms in the trinomial you are trying to factor. If it is negative, and since you are multiplying the Last terms of the binomials to arrive at it, one of the last terms needs to be positive and one negative: ( x + ) ( x - ).

Keeping in mind that these two values when added produce the a€?ba€? value of the middle term. There are only two ways this can happen, Since it is the product of the Last terms of the binomials, they are either both positive or both negative. Books will usually have you looking for factors of the second binomial, but the reality is that if this trinomial will factor, the second binomial must match the first (x + 5). Is it double the square root of the first term (x) times the square root of the second term (3)? If you have not figured it out yet, this process will only work when multiplying two binomials.

Therefore, if the a€?ba€? of the middle term is positive, the larger number must go into the first binomial.

Even in math new things come along once in a great while, and factoring these trinomials is as cool as sliced bread. Sure you could multiply 30 times (- 8), but our objective is always to make life easy for ourselves.

You should have determined that it is a quadratic binomial since it is of the second degree and has two terms. Write 3x, then open a parenthesis and determine what 3x must be multiplied by to achieve the polynomial you started with. All you need to do is to recognize these special cases and you can let your brain do all the work instead of using up pencil lead. If the a€?ba€? of the middle term is negative, the larger number must go into the second binomial.

You should notice that 2 is a common factor to all three terms, therefore you should factor it out first. Since they disappeared you could conclude that you ended up with the square of the first term less the square of the second term: 3600 - 1.

Therefore whatever the sign of the middle term, the last terms of the binomial factors will be the same.

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