Answer :
To find the zeros of the function [tex]\( f(x) = 12x^3 - 19x^2 - 45x - 18 \)[/tex], we're looking for the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 0 \)[/tex].
1. Identify the Type of Function: [tex]\( f(x) \)[/tex] is a cubic polynomial, which means it can have up to 3 real zeros.
2. Check for Rational Zeros: According to the Rational Root Theorem, any rational zero, [tex]\( \frac{p}{q} \)[/tex], of the polynomial, is a factor of the constant term (-18) over a factor of the leading coefficient (12). Therefore, possible rational zeros are factors of -18: [tex]\( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \)[/tex] over factors of 12: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
3. Perform Polynomial Division or Use Factoring Techniques: We will directly list the zeros without the detailed calculation steps:
- The zeros of the polynomial are [tex]\( x = -\frac{3}{4}, x = -\frac{2}{3}, \)[/tex] and [tex]\( x = 3 \)[/tex].
4. Verify: You can verify these solutions by plugging each into the original polynomial to check if they make the function equal zero.
Thus, the zeros of the function [tex]\( f(x) = 12x^3 - 19x^2 - 45x - 18 \)[/tex] are [tex]\(-\frac{3}{4}, -\frac{2}{3},\)[/tex] and [tex]\(3\)[/tex].
1. Identify the Type of Function: [tex]\( f(x) \)[/tex] is a cubic polynomial, which means it can have up to 3 real zeros.
2. Check for Rational Zeros: According to the Rational Root Theorem, any rational zero, [tex]\( \frac{p}{q} \)[/tex], of the polynomial, is a factor of the constant term (-18) over a factor of the leading coefficient (12). Therefore, possible rational zeros are factors of -18: [tex]\( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \)[/tex] over factors of 12: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
3. Perform Polynomial Division or Use Factoring Techniques: We will directly list the zeros without the detailed calculation steps:
- The zeros of the polynomial are [tex]\( x = -\frac{3}{4}, x = -\frac{2}{3}, \)[/tex] and [tex]\( x = 3 \)[/tex].
4. Verify: You can verify these solutions by plugging each into the original polynomial to check if they make the function equal zero.
Thus, the zeros of the function [tex]\( f(x) = 12x^3 - 19x^2 - 45x - 18 \)[/tex] are [tex]\(-\frac{3}{4}, -\frac{2}{3},\)[/tex] and [tex]\(3\)[/tex].
