from a handpicked tutor in LIVE 1-to-1 classes
Which of the following represents the zeros of f(x) = 3x3 - 10x2 - 81x + 28?
7, -4, 1/3
7, -4, -1/3
7, 4, 1/3
7, 4, -1/3
Solution:
Using the Rational Zeros Theorem, which states that, if the polynomial f(x) = anxn + an - 1xn - 1 + ... + a1x + a0 has integer coefficients,
then every rational zero of f(x) has the form p/q where p is a factor of the constant term a₀ and q is a factor of the leading coefficient an.
Given:
Function f(x) = 3x3 - 10x2 - 81x + 28
Here,
p: ±1, ±2, ±4, ±7, ±14, ±28 which are all factors of constant term 28
q: ±1, ±3 which are all factors of the leading coefficient 3
All possible values are
p/q: ±1, ±2, ±4, ±7, ±14, ±28, ±1/3, ±2/3, ±4/3, ±7/3, ±14/3, ±28/3,
From the given options we can select only ±4, ±7, ±1/3 to verify the roots.
f(x) = 3x3 - 10x2 - 81x + 28
⇒ f(4) = 3(4)3 - 10(4)2 - 81(4) + 28
= 192 - 160 - 324 + 28
= -264
⇒ f(-4) = 3(-4)3 - 10(-4)2 - 81(-4) + 28
= -192 - 160 + 324 + 28
= 0
⇒ f(7) = 3(7)3 - 10(7)2 - 81(7) + 28
= 1029 - 490 - 567 + 28
= 0
⇒ f(1/3) = 3(1/3)3 - 10(1/3)2 - 81(1/3) + 28
= 0.111 - 1.111 - 27 + 28 = 0
Therefore, the zeros of polynomial are -4, 7, 1/3.
Which of the following represents the zeros of f(x) = 3x3 - 10x2 - 81x + 28?
Summary:
The zeros of f(x) = 3x3 - 10x2 - 81x + 28 are -4, 7, 1/3.
visual curriculum