[tex]\underline{\underline{\large{\red{\mathcal{✒GIVEN:}}}}}[/tex]
[tex]\bullet \: \: \rm{a. \: 4 {x}^{2} - 21 {x}^{3} + 18 {x}^{2} + 19x - 6 = 0 }[/tex]
[tex]\bullet \: \: \rm{b. \: 2 {x}^{3} - {3x}^{2} - 10x - 4 = 0}[/tex]
[tex]\bullet \: \: \rm{c. \: {x}^{4} - {4x}^{2} + 3 = 0}[/tex]
[tex]\underline{\underline{\large{\red{\mathcal{REQUIRED:}}}}}[/tex]
Hi, Brainly User. I will help you solve the following equation. We need to use the Rational Root Theorem to find the zeros of the equations.
[tex]\underline{\underline{\large{\red{\mathcal{SOLUTION:}}}}}[/tex]
[tex]\sf{A.}[/tex]
[tex]\bm{ 4 {x}^{2} - 21 {x}^{3} + 18 {x}^{2} + 19x - 6 = 0 }[/tex]
The trailing coefficient is -6.
Factors:
[tex]\tt{\pm 1, \pm 2, \pm 3, \pm 6}[/tex]
The trailing coefficient is -6.
Factors:
[tex]\tt{ \pm 1, \pm 2, \pm 3, \pm 6}[/tex]
[tex]\tt{ \pm 1, \pm 2, \pm 4}[/tex]
These are the possible values for q.
All possible values of [tex]\rm{\dfrac{p}{q}}[/tex]:
[tex] \small{\tt{\pm \frac{1}{1} , \pm \frac{1}{2} , \pm \frac{1}{4} , \pm \frac{2}{1} , \pm \frac{2}{2} , \pm \frac{2}{4} , \pm \frac{3}{1} , \pm \frac{3}{2} , \pm \frac{3}{4} , \pm \frac{6}{1} , \pm \frac{6}{2} , }}[/tex] [tex]\small{\tt{\pm \frac{6}{4} }}[/tex]
Simplifying and removing the duplicates. These are the possible rational roots:
[tex]\tt{ \pm 1, \pm \frac{1}{2} , \pm \frac{1}{4} , \pm 2, \pm 3, \pm \frac{3}{2} , \pm \frac{3}{4} , \pm 6}[/tex]
Now, check the possible roots: if a is a root of the polynomial, the remainder from the division of the polynomial by [tex]\rm{ (x - a) }[/tex] should equal 0. By checking all possible roots, we find the actual rational roots are:
[tex]\large{\tt{\purple{2, - \dfrac{3}{4} }}}[/tex]
[tex]\sf{B.}[/tex]
[tex]\bm{ 2 {x}^{3} - {3x}^{2} - 10x - 4 = 0}[/tex]
The trailing coefficient is -4.
Factors:
[tex]\tt{ \pm 1, \pm 2, \pm 4}[/tex]
These are the possible values for q.
All possible values of [tex]\rm{\dfrac{p}{q}}[/tex]:
[tex]\tt{ \pm \frac{1}{1} , \pm \frac{1}{2} , \pm \frac{2}{1} , \pm \frac{2}{2} , \pm \frac{4}{1} , \pm \frac{4}{2} }[/tex]
Simplifying and removing the duplicates. These are the possible rational roots:
[tex]\tt{ \pm1, \pm \frac{1}{2} , \pm 2, \pm4}[/tex]
Now, check the possible roots: if a is a root of the polynomial, the remainder from the division of the polynomial by [tex]\rm{ (x - a) }[/tex] should equal 0. By checking all possible roots, we find the actual rational roots are:
[tex]\large{\tt{\purple{ - \dfrac{1}{2} }}}[/tex]
[tex]\sf{C.}[/tex]
[tex]\bm{ {x}^{4} - {4x}^{2} + 3 = 0}[/tex]
The trailing coefficient is 3.
Factors:
[tex]\tt{\pm 1, \pm 3}[/tex]
These are the possible values for p.
The leading coefficient is 1.
Factors:
[tex]\tt{\pm1}[/tex]
These are the possible values for q.
All possible values of [tex]\rm{\dfrac{p}{q}}[/tex]:
[tex]\tt{ \pm \frac{1}{1} , \pm \frac{3}{1} }[/tex]
Simplifying. These are the possible rational roots:
[tex]\tt{\pm 1, \pm 3}[/tex]
Now, check the possible roots: if a is a root of the polynomial, the remainder from the division of the polynomial by [tex]\rm{ (x - a) }[/tex] should equal 0. By checking all possible roots, we find the actual rational roots are:
[tex]\large{\tt{\purple{ 1, -1}}}[/tex]
Final Answer:
》 A.
[tex]\large{\rm{\purple{2, - \dfrac{3}{4} }}}[/tex]
》 B.
[tex]\large{\rm{\purple{ - \dfrac{1}{2} }}}[/tex]
》 C.
[tex]\large{\rm{\purple{ 1, -1}}}[/tex]
