How would I answer this? Write a polynomial of least degree with roots 1,5, and -6

Question

Write a polynomial of least degree with roots 1, 5, and -6.

Write your answer using the variable x and in standard form with a leading coefficient of 1.

Answer 1

The process for answer this question is very easy if you remember the property of zero for a polynomial function, meaning that if (x-2)=0, and you solve for x , x-2=0, adding 2 in both sides we get x=2.

In our case the problem provides the x values, x=1, x=5 and x=-6.

Therefore : P(x) = (x-1) (x-5)(x+6). Find the products of these factors(FOIL)

2) f(x)= (x^2-5x-x+5)(x+6)=(x^2-6x+5) (x+6)

3) f(x)=(x^2-6x+5)(x+6)=(x^3+6x^2-6x^2-36x+5x+30). So your answer should be F(x)=x^3-31x+30

Note: To verify your answer , graph the function in a TI-84 or demos calculator (free online) and verify the interceptions. in this case you notice that graph passing trough x=1, y=0, x=5,y=0 and x=-6, y=0.

Answer 2

When you factor a polynomial you end up with it's roots.

To find it's roots you set it equal to zero and solve for all

possible values of x.

for example, a polynomial of degree two, or second order

polynomial, (2 being the highest power of x present

in the polynomial) in the form:

ax² + bx + c = 0

has two roots,

x_1, x₂ = [-b ± √(b² - 4ac)] / 2a

which factors the above polynomial into factors:

ax² + bx + c = 0 = (x - x₁)(x - x₂)

If we have three roots we have polynomial of at least degree

and it can be factored into 3 roots:

(x - x₁)(x - x₂)(x - x₃)

in this case we have

x₁ = 1, x₂ = 5, x₃ = -6

which gives a polynomial that is the product of at least

the following three factors:

(x - 1)(x - 5)(x + 6)

which, when multiplied out yields the polynomial:

(x - 1)(x - 5)(x + 6) = (x² - x - 5x + 5)(x + 6) = (x² - 6x + 5)(x + 6)

= x³ - 6x² + 5x + 6x² - 36x + 30

= x³ - 31x + 30 answer

Answer 3

If you are given the roots, you can find the factors and then multiply the factors together to get the polynomial. For the roots provided, the factors would be:

f(x) = (x-1)(x-5)(x+6) (The signs are opposite because if you set these factors equal to zero and solve, you will get the roots provided)

Now multiply. It doesn't matter which terms you do in which order. You can start with FOIL for the first set of binomials.

f(x) = (x²-x-5x+5)(x+6) Combine like terms.

f(x) = (x²-6x+5)(x+6) Now multiply this using the distributive property distributing (x+6) to each term

f(x) = (x+6) (x²)+(x+6)(-6x)+(x+6)(5) Now distribute again.

f(x) = (x²)(x)+(x²)(6)+(-6x)(x)+(-6x)(6)+(5)(x)+(5)(6) Multiply

f(x) = x³+6x²-6x²-36x+5x+30 Combine like terms

f(x) = x³-31x+30