Math Problem

Question

Determine the degree and leading coefficient for the following polynomials:

 

P(x)= -2(x-3)(x-1)^2(x^2+2)

and

p(x)= (-2x+3)^2(3x-1)+2

Answer 1

The first polynomial is P(x) = -2 (x-3) (x-1)² (x² + 2)

 

Let's expand this.  My preference is to first expand the (x-1)² term

P(x) = -2 (x-3) (x²-2x+1) (x² - 1)

 

Now let's work this from left to right

P(x) = (-2x+6)(x²-2x+1)(x²-1)

P(x) = (-2x³ +4x² -2x +6x² - 12x + 6)(x²-1)

P(x) = (-2x³ + 10x² - 14x + 6)(x² - 1)

P(x) = -2x⁵ + 2x³ + 10x⁴ - 10x² -14x³ + 14x +  6x² - 6

P(x) = -2x⁵ + 10x⁴ -12x³ - 4x² + 14x -6

So this is a 5th degree polynomial with a leading coefficient of -2

 

Regarding the 2nd polynomial, I believe there is a typo in your question.

I believe you meant to write

p(x) = (-2x+3)²(3x-1)²

Notice I've squared the second term rather than the +2 you have in your question.

 

Working with this, I'm going to square each of the two terms.

p(x) = (4x² -12x + 9)(9x² - 6x + 1)

 

Now multiply these two quadratics

p(x) = 36x⁴ - 24x³ + 4x² - 108x³ + 72x² - 12x + 81x² - 54x + 9

 

Finally combining and rearranging terms

p(x) = 36x⁴ - 132x³ + 157x² - 66x + 9

This is a 4th degree polynomial with a leading coefficient of 36.

 

 

 

Answer 2

The degree is highest exponent of x.  To find the leading coefficient, we need to expand the polynomial. 

 

p(x) = -2(x - 3)(x - 1)²(x² + 2)

 

p(x) = -2(x - 3)(x - 1)(x - 1)(x² + 2)

 

p(x) = (x² - 2x + 1)(-2x + 6)(x² + 2)

 

p(x) = (x² - 2x + 1)(-2x³ + 6x² - 4x + 12)

 

p(x) = x²(-2x³ + 6x² - 4x + 12) - 2x(-2x³ + 6x² - 4x + 12) -2x³ + 6x² - 4x + 12

 

Now just distribute the terms to get rid of parentheses.  Once you do that, combine like terms.  But wait.  You won't have to do all of that.  By looking at this expression, you will be able to see that the degree of the polynomial is 5 and the leading coefficient is -2.

 

 

Try the last one on your own.