b According to the Remainder Theorem, if p(c)=0 then (x-c) is a factor. From Part A, we know that p(2)=0. Therefore, according to the theorem, (x-2) must be a factor.
c Since the graph has a factor of (x-2), we can write the following equation.
(x-2) (other factor)=x^3-6x^2+7x+2
Let's use an area model to find the other factor. The first term of the other factor must be x^2. It must be this in order for the product of the factor's first terms to equal x^3.
( x-2) ( x^2......)= x^3-6x^2+7x+2
With this information, we can begin creating our area model's first column.
In the original expression, we have the term -6x^2. Since one tile of our area model contains - 2x^2, we must add - 4x^2 to get a sum of -6x^2. With this information, we can identify the second term of our factor and the contents of the area model's second column.
Again, examining the original expression we see the term 7x. Since one tile of our area model contains 8x, we must add - x to get a sum of 7x. With this information, we can identify the third term of our factor and the contents of the third column.
If we add all of the terms contained within the area model, the sum should equal the expression.
x^3+(- 2x^2)+(-4x^2)+8x+(- x)+2
⇓
x^3-6x^2+7x+2
Now we know that another factor is (x^2-4x-1). With this information, we can rewrite the original equation.
(x-2) (x^2-4x-1)=x^3-6x^2+7x+2
d We already know one zero is at x=2. To find the remaining zeroes, we should set the second degree polynomial we found in Part A equal to 0 and solve for x.
