Question 17
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials (iv) t3−2t2−15t.
Solution:
To find the zeroes of the polynomial t3−2t2−15t using the factorization method and verify the relations between the zeroes and the coefficients, we can follow these steps:
Step 1: Write the polynomial
We start with the polynomial:
t3−2t2−15t=0
Step 2: Factor out the common term
Notice that all terms in the polynomial have a common factor of t. We can factor t out:
t(t2−2t−15)=0
Step 3: Factor the quadratic expression
Next, we need to factor the quadratic t2−2t−15. We look for two numbers that multiply to −15 (the constant term) and add to −2 (the coefficient of t). The numbers −5 and 3 satisfy these conditions:
t2−2t−15=(t−5)(t+3)
Step 4: Write the complete factorization
Now we can write the complete factorization of the polynomial:
t(t−5)(t+3)=0
Step 5: Set each factor to zero
To find the zeroes, we set each factor equal to zero:
1. t=0
2. t−5=0 → t=5
3. t+3=0 → t=−3
Thus, the zeroes of the polynomial are:
t=0,t=5,t=−3
Step 6: Verify the relations between the zeroes and coefficients
Let the zeroes be α=0, β=5, and γ=−3.
1. Sum of the zeroes:
α+β+γ=0+5−3=2
According to the relation, this should equal −b/a:
−(−2)/1=2
2. Sum of the products of the zeroes taken two at a time:
αβ+βγ+γα=(0)(5)+(5)(−3)+(−3)(0)=0−15+0=−15
According to the relation, this should equal c/a:
−15/1=−15
3. Product of the zeroes:
αβγ=(0)(5)(−3)=0
According to the relation, this should equal −d/a:
−0/1=0
Conclusion
The zeroes of the polynomial t3−2t2−15t are 0,5,−3. The relations between the zeroes and the coefficients have been verified successfully.