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factor 2x^2-12x+16 completely

1.)factor 2x^2-12x+16
2.)Find the degree of the polynomial 9y^4-6y^3+8 and determine whether its a monomial, binomial, trinmoial, or none.
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2 Answers

The question says factor completely so actually the correct answer is 2(x - 2)(x - 4).
So you see the question would have been much easier if the 2 was factored out first because there would not have been so many choices.
First factor out 2 to get 2(x2 - 6x + 8)
Now the factoring is easy. We need two numbers whose product is +8 and whose sum is -6. If a product is positive but the sum is negative, the two numbers must both be negative.
Therefore we quickly get    2(x - 2)(x - 4)
Good evening, Shayna!

For the first question, to factor the polynomial, we kinda sorta have to use FOIL backwards. You probably already know that when you have two binomials multiplied, you sort it out by multiplying the (F)irst terms, then the (O)uter terms, then the (I)nner terms, then the (L)ast terms.

So, let's dissect 2x^2-12x+16. For the first terms, what can we multiply to get 2x^2? 2x and x. So, now we know we have something where 2x - or + something will be multiplied by x - or + something else, or
(2x-/+a)(x-/+b). From here, let's figure out what the outer terms have to be to get a positive 16. Think of all the factors that will give us 16. There are: 1 and 16, 2 and 8, 4 and 4, -1 and -16, -2 and -8, and -4 and -4. Note that the middle term is a negative, so most likely we are going to have to use one of the negative groups of factors for this.

Let's plug stuff in. (2x-1)(x-16) doesn't work. It gives us 2x^2-32x-x+16 => 2x^2-33x+16. That's definitely not it. With -2,-8, we get (2x-2)(x-8) => 2x^2-16x-2x+16 => 2x^2-18x+16. Nope, not it. With -4,-4, we have (2x-4)(x-4) => 2x^2-8x-4x+16 => 2x^2-12x+16. Bingo. So, 2x^2-12x+16 factored is (2x-4)(x-4).

For the second question, to determine whether it is a monomial, binomial, or trinomial, all you have to do is count the number of individual terms in the expression. Count them by noting how many are separated by + or -. For example, an expression such as 4xy is a monomial (only has one term). An expression such as 4xy-1 has 2 terms, 4xy and 1, thus it is a binomial. In your second question, there are three terms: 9y^4, 6y^3, and 8, and so it is a trinomial.  And, to determine its degree, simply look at the power of the highest-powered term.  In your case, the highest-powered term is 9y^4, so its degree is 4.

Hope this helps!