Construct a polynomial function with the following properties

Question

fifth degree, 1 is a zero of multiplicity 2,  -2 is the only other zero, leading coefficient is 3.

Answer 1

Greetings! Lets solve this shall we ?

So, we must find a polynomial of degree 5 with a zero of 1 and a multiplicity of 2 and -2 is the second and last zero. -2 is the last zero, hence the factor of the polynomial is

x³+8=0

x³ = -8..............Cube root both sides such that

x = -2

3 is the leading coefficient such that

3(x - 1)²(x³+8)..............Then we multiply everything out such that

3 (x²-2x+1)(x³+8)............Then

3(x⁵+8x²-2x⁴-16x+x³+8)...........Finally,

= 3x⁵-6x⁴+3x³+24x²-48x+24

I hope this helped!

Answer 2

A fifth degree polynomial means that the highest number exponent on our variable is 5. Since 1 is a zero with a multiplicity of two, we know that (x-1) is one of our factors, and a "multiplicity of two" means we use it twice to write the factored form of the final polynomial. Therefore, part of our polynomial looks like this:

y = (x-1)²

Likewise, if -2 is the other zero, the corresponding factor is (x+2).

(x-1)² gives us a leading term of x², but we need a fifth degree term, with x⁵. So we'll use (x+2)³ for the next part.

Given the leading coefficient of 3, our polynomial is now:

y = 3(x-1)²(x+2)³

Square that first binomial to get:

y = 3(x²-2x+1)(x+2)³.

And cube the second binomial to get:

y = 3(x²-2x+1)(x³+6x²+12x+8)

Multiply the two polynomials to get:

y = 3(x⁵+4x⁴+x³-10x²-4x+8)

And finally, multiply everything by three so the leading coefficient is 3:

y = 3x⁵+12x⁴+3x³-30x²-12x+24

...and that's it!

I hope this helps! Also, if you're having trouble with concepts, check with your teacher, or on Khan Academy, or get in touch! The hardest thing about a question like this should be addition and subtraction, not the concepts.