NCERT Solutions for Class 9 Maths Exercise 2.3 Question 3
Understanding the Question 🧐
To check if one polynomial is a factor of another, we can use the Factor Theorem. This theorem is a special case of the Remainder Theorem.
So, our goal is to find the remainder. If the remainder is 0, it’s a factor. If not, it isn’t.
Check whether &&7 + 3x&& is a factor of &&3x^3 + 7x&&.
Applying the Factor Theorem 📝
Step 1: Identify the polynomials.
The polynomial (dividend) is &&p(x) = 3x^3 + 7x&&.
The potential factor (divisor) is &&g(x) = 7 + 3x&&.
Step 2: Find the zero of the potential factor &&g(x)&&.
To do this, we set &&g(x) = 0&& and solve for &&x&&:
&&7 + 3x = 0&&
&&3x = -7&&
&&x = -\frac{7}{3}&&
Step 3: Substitute this zero into the main polynomial &&p(x)&& to find the remainder.
We need to calculate &&p(-\frac{7}{3})&&:
&&p(-\frac{7}{3}) = 3(-\frac{7}{3})^3 + 7(-\frac{7}{3})&&
Step 4: Simplify the expression.
&&= 3(-\frac{7 \times 7 \times 7}{3 \times 3 \times 3}) – \frac{49}{3}&&
&&= 3(-\frac{343}{27}) – \frac{49}{3}&&
We can cancel a 3 from the numerator and denominator:
&&= -\frac{343}{9} – \frac{49}{3}&&
To subtract, we find a common denominator (9):
&&= -\frac{343}{9} – \frac{49 \times 3}{3 \times 3}&&
&&= -\frac{343}{9} – \frac{147}{9}&&
&&= \frac{-343 – 147}{9} = \frac{-490}{9}&&
Step 5: Check if the remainder is zero.
The remainder is &&-\frac{490}{9}&&, which is not equal to 0.
Conclusion: Since the remainder is not 0, &&7 + 3x&& is not a factor of &&3x^3 + 7x&&.
FAQ ❓
Q: What is the Factor Theorem?
A: The Factor Theorem says that a linear polynomial &&(x – a)&& is a factor of another polynomial &&p(x)&& if and only if &&p(a) = 0&&. It’s a quick way to check for factors without doing long division.
Q: How is the Factor Theorem related to the Remainder Theorem?
A: The Factor Theorem is a special outcome of the Remainder Theorem. The Remainder Theorem finds the remainder (&&p(a)&&). The Factor Theorem just checks if that specific remainder is zero.
Further Reading 📖
To learn more about the Factor Theorem and finding factors of polynomials, you can refer to the official NCERT textbook for Class 9 Maths, Chapter 2. More resources are available on the NCERT website at https://ncert.nic.in/.
