Verified answer

Answer:

[tex]\qquad\qquad\boxed{ \sf{ \: \bf \: Remainder= \: \dfrac{31}{8} \: }} \\ \\ [/tex]

Step-by-step explanation:

Given polynomial is

[tex]\sf \: {x}^{3} + {5x}^{2} + 3x + 1 \\ \\ [/tex]

Let assume that

[tex]\sf \: f(x) = {x}^{3} + {5x}^{2} + 3x + 1 \\ \\ [/tex]

Now, we have to find the remainder when f(x) is divided by

[tex]\sf \: x - \dfrac{1}{2} \\ \\ [/tex]

We know,

Remainder Theorem:-

This theorem states that if a polynomial f(x) of degree greater than or equals to one is divided by linear polynomial x - a, then remainder is f(a).

So,

[tex]\sf \: Remainder = f\left(\dfrac{1}{2} \right) \\ \\ [/tex]

[tex]\sf \: = \: {\left(\dfrac{1}{2} \right)}^{3} + 5 {\left(\dfrac{1}{2} \right)}^{2} + 3\left(\dfrac{1}{2} \right) + 1 \\ \\ [/tex]

[tex]\sf \: = \: \dfrac{1}{8} + \dfrac{5}{4} + \dfrac{3}{2} + 1 \\ \\ [/tex]

[tex]\sf \: = \: \dfrac{1 + 10 + 12 + 8}{8} \\ \\ [/tex]

[tex]\sf \: = \: \dfrac{31}{8} \\ \\ [/tex]

Hence,

[tex]\sf \: \sf\implies \bf \: Remainder= \: \dfrac{31}{8} \\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

ADDITIONAL INFORMATION

[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]

Here, given a cubic polynomial, x³ + 5x² + 3x + 1 which is divided by x - ½.

We know that remainder theorem states that when we divide a polynomial f(x) by (x - c), then the remainder is f(c)

So, here given polynomial is x³ + 5x² + 3x + 1 and divisor is x - ½

=> x - ½ = 0

=> x = 0 + ½

=> x = ½

Now,

=> f(x) = x³ + 5x² + 3x + 1

=> f(½) = (½)³ + 5(½)² + 3(½) + 1

=> f(½) = ⅛ + 5/4 + 3/2 + 1

=> f(½) = ⅛ + 5/4 + 3/2 + 1

=> f(½) = (1 + 10 + 12 + 8)/8

=> f(½) = 31/8

∴ Hence, the remainder is 31/8.