Confirm Degree & Dominant Term of Polynomial Equation

[dylanjames]

Can someone just confirm my answers to this easy polynomial question,
State the degree and dominant term to f(x)=2x(x-3)^3(x-1)(4x-2)

I am working on this online and there is nothing on working on equations like this in the lesson. I believe the degree to be either 2 or 6, as the functions end behavior models an even number.
The dominant term in that case would be 2x^2 or 2x^6...

any clarification appreciated.

 

[Mark44]

Can someone just confirm my answers to this easy polynomial question,
State the degree and dominant term to f(x)=2x(x-3)^3(x-1)(4x-2)

I am working on this online and there is nothing on working on equations like this in the lesson. I believe the degree to be either 2 or 6, as the functions end behavior models an even number.

It's very easy to determine that this is not a 2nd degree polynomial -- just expand the right side of your equation. I don't know why you would think this would be a 2nd degree polynomial.

The dominant term in that case would be 2x^2 or 2x^6...

No. One of your factors is 4x - 2. That 4 will show up as part of your dominant term.

 

[dylanjames]

I can't expand that to save my life.

 

[Mark44]

I can't expand that to save my life.

Can you multiply x - 1 and 4x - 2?

The way you multiply a product of several factors is to multiply two of them together, and then multiply that by one of the other factors, and keep going until you're all done.

 

[dylanjames]

So you're saying to multiply (x+1)(4x-2) and then by the first factor? I can expand in general but not sure what to do with the 2x(x-3)^3 bit.
Multiplying the (x+1)(4x-2) you would obviously get 4x^2+2x-2..so how do you proceed from there?

 

[dylanjames]

Apparently it was my original equation that I mis-typed. The true equation is 2x(x-3)^3(x+1)(4x-2)

 

[Mark44]

So you're saying to multiply (x+1)(4x-2) and then by the first factor? I can expand in general but not sure what to do with the 2x(x-3)^3 bit.
Multiplying the (x+1)(4x-2) you would obviously get 4x^2+2x-2..so how do you proceed from there?

Then multiply this by 2x.
Next, expand (x - 3)³, which will give you x³ + some lower degree terms.

Apparently it was my original equation that I mis-typed. The true equation is 2x(x-3)^3(x+1)(4x-2)

That won't make any difference as to the degree of the polynomial.

You don't have to actually multiply everything out -- just figure out what is the highest power of x and what its coefficient is.

It works in a similar way as multiplication of numbers. Suppose you had 23 * 138 * 69. This is (2*10 + 3)(1*10² + 3*10 + 8)(6*10 + 9). If I multiply only the highest degree terms in each expression, I get (2*10)(1*10²)(6*10), or (2 * 1 * 6)10⁴, which is 12 x 10,000 = 120,000.

So if I were to actually multiply 23 times 138 time 69, I should get an answer around 120,000, but a bit larger than this, since I didn't carry out the multiplication completely.

 

[dylanjames]

Im not sure if I am on the right track here or not..

f(x)=2x(x-3)^3(x+1)(4x-2)
=2x(x-3)^3(4x^2-2x-2+4x)
=2x^(2*3)-6x^3(8x^2-4x-4+8x)

Did I solve this out ok? If this were the case would 6 not be the degree and 2x^3 the dominant figure?

 

[HallsofIvy]

If your question is just "what is the degree?" and "what is the leading coefficient?", you don't have do the entire multiplication. The "leading term" in each factor is, in order, 2x, x^3, x, and 4x. Multiplying those together, we have (2x)(x^3)(x)(4x)= (4)(2)x^(1+ 3+ 1+ 1)= 8x^6 which is the leading term of 2x(x- 3)^3(4x- 2). The degree is 6 and the leading coefficient is 8.

 

[dylanjames]

HALLELUJAH! Sir you are one brilliant individual. I could not figure out for the life of me why there was no material in the text on expanding out equations like this, and much less for two marks. CHEERS mate.