Answer:
- <u>Below it is shown the procedure and that +3 is one rational zero.</u>
Explanation:
The function is:
According to the rational roots theorem, the possible rational roots of a polynomial are the negative and positive quotients of the factors of the constant term (45 in this case) divided by the factors of the coefficient of the highest degree term (7 in this case).
Then, the possible rational roots are the divisors of 45 divided by the divisors of 7.
- Divisors of 45: ±1, ± 3; ± 5, ±9, ±15, ± 45
- Divisors of 7: ±1, ±7
Then, the possible rational roots are:
- ±1, ±1/7, ±3, ±3/7, ±5, ±5/7, ±9, ±9/15, ±15, ±15/7, ±45, ±45/7
Now you should test them to find out whether they are factors or not.
You need to find just one ration zero. Thus , I will show you the procedure, with one of them: use 3.
Then, you have to divide the original polynomial by x - 3.
The sythetic division is:
<u>1. Write the root and the coefficients in this form:</u>
+3 | 7 - 5 - 63 +45
|
=========================
<u>2. Copy the first coefficient, 7, below the horizontal line:</u>
+3 | 7 - 5 - 63 +45
|
=========================
7
<u />
<u>3. Multiply the root (3) by 7, write the product below the second coefficient, and find the sum: 3 × 7 = 21; 21 + (-5) = 16</u>
+3 | 7 - 5 - 63 +45
| 21
=========================
7 16
<u>4. Multiply the root (3) by 16, write the product below the next coefficient, and find the sum: 3 × 16 = 48; 48 + (-63) = - 15</u>
+3 | 7 - 5 - 63 +45
| 21 +48
=========================
7 16 -15
<u />
<u>5. Multiply the root (3) by -15, write the product below the next coefficient, and find the sum: 3 × (-15)= - 45; -45 + 45 = 0</u>
+3 | 7 - 5 - 63 +45
| 21 +48 -45
=========================
7 16 -15 0
Since the final total, which is the remainder of the division, is zero, +3 is a rational root of the polynomial.
You can now test others, to find if they are factors too.