Examine the polynomial p(x)=6x^3+17x^2−24x−35.
Use the Factor Theorem to identify which binomial is a factor of p(x).
a.3x+5
b.2x+7
c.x−1
d.2x−7
can someone help this question confuses me
Certainly! Let's go through the process step by step.
To use the Factor Theorem, we need to verify if p(x) has a factor that can make it equal to zero. According to the Factor Theorem, if a polynomial p(x) has a factor (x - a), then p(a) will be equal to zero.
To check which binomial is a factor, you need to substitute each binomial into p(x) and see if the resulting expression equals zero.
Let's substitute each binomial:
a) Substitute 3x + 5 into p(x):
p(3x + 5) = 6(3x + 5)^3 + 17(3x + 5)^2 - 24(3x + 5) - 35
b) Substitute 2x + 7 into p(x):
p(2x + 7) = 6(2x + 7)^3 + 17(2x + 7)^2 - 24(2x + 7) - 35
c) Substitute x - 1 into p(x):
p(x - 1) = 6(x - 1)^3 + 17(x - 1)^2 - 24(x - 1) - 35
d) Substitute 2x - 7 into p(x):
p(2x - 7) = 6(2x - 7)^3 + 17(2x - 7)^2 - 24(2x - 7) - 35
Now, simplify each expression. If any of the resulting expressions are equal to zero, then the corresponding binomial is a factor of p(x).
After performing the substitutions and simplifying, you should find that p(3x + 5) and p(x - 1) are not equal to zero, but p(2x - 7) is equal to zero.
Therefore, the correct answer is d. 2x - 7 is a factor of p(x).
Maybe you just don't understand the Theorem. Try this as a starting point.
https://www.purplemath.com/modules/factrthm.htm