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
I thought that 2x+7 was a factor of p(x) but I don't know and I am confused now can somebody help
Ok, let's test that one
If 2x + 7 is a factor, then f(-7/2) should be zero.
f(x) = 6x^3+17x^2−24x−35
f(-7/2) = 6(-7/2)^3 + 17(-7/2)^2 - 24(-7/2) - 35
= 6(-343/8) + 17(49/4) + 168/2 - 35
= -2058/8 + 1666/8 + 672/8 - 280/8 = 0/8 = 0
Yeah, so 2x+7 is a factor
try the others in the same way
To determine which binomial is a factor of the polynomial using the Factor Theorem, you need to use synthetic division. The Factor Theorem states that if a polynomial f(x) has a factor (x - c), then f(c) = 0.
To begin, you would set up a synthetic division using the possible factor (binomial) and the coefficients of the polynomial. In this case, we will use option b: 2x + 7. To do this:
Step 1: Write the polynomial coefficients in descending order and leave a placeholder for missing power terms.
p(x) = 6x^3 + 17x^2 - 24x - 35
Step 2: Set up the synthetic division as follows:
-7 | 6 17 -24 -35
--------------------
Step 3: Begin the synthetic division by bringing down the first coefficient, which is 6.
-7 | 6 17 -24 -35
--------------------
6
Step 4: Multiply the value just written (6) by the divisor (2x + 7) and write the result under the next coefficient.
-7 | 6 17 -24 -35
--------------------
6
0
Step 5: Add the two values in the next column.
-7 | 6 17 -24 -35
--------------------
6
0
_______
17
Step 6: Repeat steps 4 and 5 until you have reached the final column.
-7 | 6 17 -24 -35
--------------------
6 0
_______
17 -7
_______
0
Step 7: The last value in the synthetic division result should be zero. If it is zero, then the binomial (2x + 7) is indeed a factor of the original polynomial (p(x)). In this case, the remainder is zero, indicating that (2x + 7) is a factor of the polynomial p(x).
Therefore, the correct answer is option b: (2x + 7) is a factor of p(x).