b Is there a greatest common factor (GCF) between all of the terms in the given expression? If so, you should factor that out first.
C
c Use the Zero Product Property.
A
a x=4/3 or x=1
B
b x=0 or x=- 6
C
c x=- 5 or x=3/2
Practice makes perfect
a To find where a function intercepts the x-axis, the function can be set equal to zero. Then, the x-values that satisfy the equation are the zeros of the function, also called the roots. Thus, to find the roots of the given polynomial, we can set it equal to zero and solve for x.
3x^2-7x+4=0
We want to solve the above equation for x.
To do this, we can start by factoring. Then, we will use the Zero Product Property.
Factoring
Here we have a quadratic trinomial of the form ax^2+bx+c, where |a| ≠1 and there are no common factors. To factor this equation, we will rewrite the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b.
3x^2-7x+4=0
⇕
3x^2+(- 7)x+4=0
We have that a= 3, b=- 7, and c=4. There are now three steps we need to follow in order to rewrite the above equation.
Find a c. Since we have that a= 3 and c=4, the value of a c is 3* 4=12.
Find factors of a c. Since a c=12, which is positive, we need factors of a c to have the same sign — both positive or both negative — in order for the product to be positive. Since b=- 7, which is negative, those factors will need to be negative so that their sum is negative.
Rewrite bx as two terms. Now that we know which factors are the ones to be used, we can rewrite bx as two terms.
3x^2+(- 7)x+4=0
⇕
3x^2 - 3x - 4x+4=0
Finally, we will factor the last equation obtained.
b To find where a function intercepts the x-axis, the function can be set equal to zero. Then, the x-values that satisfy the equation are the zeros of the function, also called the roots. Thus, to find the roots of the given polynomial, we can set it equal to zero and solve for x.
x^2+6x=0We want to solve the above equation for x.
To do this, we can use factoring. We will start from identifying the greatest common factor (GCF). Then, we will use the Zero Product Property.
Factor Out the GCF
The GCF of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. In this case, the GCF is x.
c To find where a function intercepts the x-axis, the function can be set equal to zero. Then, the x-values that satisfy the equation are the zeros of the function, also called the roots. Thus, to find the roots of the given polynomial, we can set it equal to zero and solve for x.
(x+5)(- 2x+3)=0
Since the equation is already written in factored form, we will use the Zero Product Property.
