a Where does the graph intercept the x-axis on one of the ticks?
B
b Substitute your answer from Part A into the original equation.
C
c Substituting the solution into the related factor should make it equal 0.
D
d Use an area model.
E
e The new factor is the answer from Part D.
F
f The other factor is a second degree polynomial. Set it equal to zero and solve for x.
A
a x=-7
B
b The solution is correct.
C
c (x+7)
D
d (x^2-2x-2)
E
e See solution.
F
f x_1=-7
x_2=1+sqrt(3)
x_3=1-sqrt(3)
Practice makes perfect
a The equation will have an integer solution where the graph intersects the x-axis in an integer. From the graph we see that this happens at x=-7, which is the integer solution.
b If x=-7 is a solution, we should be able to substitute it into the equation and get y=0.
Since y=0, we know that x=-7 is an integer solution.
c Since x=-7 is a solution, substituting x=-7 into the related factor should make it equal to 0. Therefore, the factor must be (x+7).
d Let's use an area model to factor the polynomial. We know that x+7 multiplied by another factor should equal x^3+5x^2-16x-14. With this information, we know that the first variable in our second 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+7)( x^2......)= x^3+5x^2-16x-14
With this information, we can begin creating our area model.
Notice that the original equation contains the term 5x^2. Since one tile of our area model contains 7x^2, we must add -2x^2 to get a sum of 5x^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 equation, we see that it contains - 16x. Since one tile of our area model contains - 14x, we must add -2x to get a sum of - 16x. With this information, we can identify the third term of our factor and the contents of the third column.
If we add the terms contained within the area model, the sum should equal the expression on the right-hand side of the original equation.
x^3+7x^2+(- 2x^2)+(- 14x)+(- 2x)+(- 14)
⇓
x^2+5x^2-16x-14
The second factor is x^2-2x-2.
e From Part D, we know that the second factor is x^2-2x-2. With this information we can complete the equation.
x^3+5x^2-16x-14=(x+7)( x^2-2x-2)=0
f By solving the equation x^2-2x-2=0, we can find the second and third solution to the original equation. We can, for example, complete the square.
