b Divide x^3-x^2-7x+3 with the factor you identified in Part A.
A
a See solution.
B
b x_(1,2)=3
x_3=-1+sqrt(2)
x_4=-1-sqrt(2)
Practice makes perfect
a We know that (x-3) is a factor, which means the graph intercepts the x-axis at x=3. However, examining the graph, we notice that the graph just touches the x-axis at x=3. This means we have a double root here.
Therefore, we know that two factors will be (x-3), which means we can write the following equation.
(x-3) (other factor)=x^3-x^2-7x+3
b Let's use an area model to find the remaining factor. The first term of the other factor must be x^2. It has to be this in order for the product of the factor's first terms to equal x^3.
( x-3) ( x^2......)= x^3-x^2-7x+3
Now we can begin creating our area model's first column.
In the original expression, we have the term - x^2. Since one tile of our area model contains - 3x^2, we must add 2x^2 to get a sum of - x^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 - 6x, 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+(- 3x^2)+2x^2+(- 6x)+(- x)+3
⇓
x^3-x^2-7x+3
Now we know that the other factor is (x^2+2x-1). With this information, we can rewrite the original expression.
(x-3) (x^2+2x-1)=x^3-x^2-7x+3
We can find the remaining roots by equating x^2+2x-1 with 0 and solving for x by using the Quadratic Formula.
The two remaining roots are x=-1+sqrt(2) and x=-1-sqrt(2), which means their corresponding factors are (x-(-1+sqrt(2))) and (x-(-1-sqrt(2))). Now we can also fully factor p(x).
p(x)=(x-3)^2(x-(-1+sqrt(2)))(x-(-1-sqrt(2)))
