Question

Obtain all the zeroes of 2x⁴ - 7x³ - 13x² + 63x - 45 , if two of its zeroes are 1 and 3.
3 and -5/2
-3 and 5/2
-3 and 5/2
-3 and -5/2​

Answer 1

Solution :

$\sf{We\:have\:polynomial\:f(x)=2x^{4} -7x^{3} -13x^{2} +63x-45}\\ \sf{Their \:zeroes\:are\:1\: \&\:3.}$

The factor of the polynomial which divided by given polynomial f(x) :

⇒ (x-1)(x-3)

⇒ x² -3x -1x + 3

⇒ x² - 4x + 3

Now;

$\boxed{\begin{array}{l|n|r}\sf x^{2} -4x+3 & \sf 2x^{4} -7x^{3} -13x^{2} +63x - 45 &\sf 2x^{2} +x-15\\ & \sf 2x^{4} -8x^{3} +6x^{2} \\ & (-)\:\:(+)\:\:(-) \\ &\rule{100}{0.8}\\ &\sf \qquad\quad x^{3} -19x^{2} +63x \\ & \sf\qquad\quad x^{3} -4x^{2} +3x\\ &\qquad \:\:(-) \:\:(+)\:\:(-)\\ &\qquad\quad \rule{70}{0.8}\\ &\sf \qquad\qquad\sf \:-15x^{2} \:+60x\:-45\\ &\qquad\qquad \sf \:-15x^{2}\: +60x\:-45 \\ & \qquad \qquad \:\:(+)\:\:(-)\:\:(+)\\ &\qquad \rule{100}{0.8} \\ & \sf \qquad\qquad 0\end{array}}$

$\bigstar$We get other factor f(x) = 2x² + x - 15

Zero of the polynomial f(x) = 0

$\longrightarrow\sf{2x^{2} +x-15=0}\\\\\longrightarrow\sf{2x^{2} +6x-5x-15=0}\\\\\longrightarrow\sf{2x(x+3)-5(x+3)=0}\\\\\longrightarrow\sf{(x+3)(2x-5)=0}\\\\\longrightarrow\sf{x+3=0\:\:\:Or\:\:\:2x-5=0}\\\\\longrightarrow\sf{x=-3\:\:\:Or\:\:\:2x=5}\\\\\longrightarrow\bf{x=-3\:\:\:Or\:\:\:x=5/2}$

Thus;

All the zeroes of the polynomials are 1,3,-3 & 5/2 .

Answer 2

Step-by-step explanation:

zeroes are 1,3,-3 and 5/2